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UNIVERSITYTDF  CALIFORNIA. 

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FIELD    MANUAL 


FOR 


ENGINEERS. 


BY 

PHILETUS    H.    PHILBRICK,   C.E.,   M.S. 

M.   AM.  MATH.   Soc., 
Chief  Engineer,  Kansas  City,   Wat\ins  and  Gulf  Railway, 

.A'.  Am.  Land  and  Timber  Co.,  etc.,  etc.; 

Sometime  Professor  of  Civil  Engineering  at  the 

State  University  of  Iowa. 


FIRST   EDITION. 
FIRST   THOUSAND. 

: 


extended,    and    extra    topi*, 
,10r  clearness,  considered  in  connection  with  the 
icrai  matter  of  the  text.  ..xpi<. 

Table  II  dispenses  with  all  calculation  Dy  numerous  readings 

the  last  figure. 
\j  &  4-  .L>aces,  has  bden  in  prepara- 


IV  PREFACE. 

as  shown  in  Chapter  IV;  Table  VII  greatly  simplifies  the 
finding  of  the  tangent  and  the  external  of  any  curve;  and 
Tables  XVIII  and  XIX,  and  some  others  to  which  the  remark 
can  apply,  are,  it  is  believed,  in  terms  of  the  proper  arguments 
and  in  the  best  form. 

In  Chapter  IV  the  laws  of  errors  in  field-work  are  demon- 
strated and  illustrated;  and  the  best  method  of  conducting  a 
preliminary  survey,  introduced  by  the  author  a  generation  ago, 
is  explained. 

In  Chapter  V  simple  and  exact  formulas  for  determining  the 
height  of  a  mountain  or  other  object  by  the  dip  of  the  horizon 
are  substituted  in  place  of  the  approximate  formulas  in  use. 

For  the  stadia,  as  well  as  for  the  telemeter,  new  formulas  are 
found  which  require  no  general  computation;  and  the  formula 
for  finding  the  proper  elevation  on  curves,  unlike  other  formulas, 
involves  no  large  factors. 

In  addition  to  a  very  general  treatment  of  Compound  Curves, 
Chapter  VI  includes  the  location  of  such  curves  of  any  number 
of  branches  (pp.  143-5),  as  well  as  easy  and  symmetrical 
formulas  for  finding  their  tangents  (p.  176). 

The  reader  interested  in  the  philosophy  of  mathematics  will 
find,  it  is  hoped,  an  elegant  and  fruitful  illustration  of  the  prin- 
ciples of  substitutions  in  determining  general  curves  to  fulfill 
required  conditions  on  pages  145,  146.  This  is  susceptible  of 
general  application.  There  has  also  been  added  a  general  treat- 
ment of  the  subject  of  Curves  tangent  to  Curves,  including  the 
"Wye  Problems";  and  also  that  of  Concentric  Curves  applic- 
able to  Parallel  Turnouts. 

The  finding  of  the  angles  between  the  rails  at  the  crossings 
of  curved  tracks  is  also  thought  to  be  a  valuable  addition. 

Chapter  VII  includes  a  variety  of  problems  in  Reversed  Curves. 
The  solution  of  Problems  IV,  V,  and  VI  were  first  given  to  the 
Senior  Class  (1869)  of  the  University  of  Michigan,  while  the 
author  was  in  temporary  charge. 

Chapter  VIII  treats  extensively  of  Turnouts.    The  distinction 

between  connecting  a  straight  line  and  a  track,  and  two  tracks, 

and  what  \^-^  iuired  in  each  case,  is  shown  on  pages  199,  200. 

of    certain    proposed    turnout    curves    is 

and  a  variety  of  methods  of  laying  out 


PREFACE.  V 

Chapter  IX  applies  entirely  to  the  author's  True  Transition 
Curve. 

Chapter  X  shows  that  the  formulas  for  the  computation  of 
earthwork  may  be  abridged  one  half  by  supposing  that  the 
side  slopes  are  produced  in  this  intersection.  A  general  relation 
between  the  "  end-area  "  volume  and  the  "  middle-area  "  volume 
is  shown  by  means  of  symbols,  and  new  formulas  are  given  for 
the  volume  of  a  frustrum  of  a  pyramid  and  the  frustrum  of  a 
cone.  A  formula  showing  the  true  correction  of  earthwork  for 
curvature  is  also  deduced,  and  the  best  method  of  computing 
earthwork  tables  explained  and  illustrated. 

Chapter  XI  contains  only  a  brief  exposition  of  the  subject 
of  approximate  and  abridged  computations,  which  it  is  thought 
may  serve  to  encourage  the  shortening  of  computations. 

Chapter  XII  describes  the  processes  incident  to  construction 
and  calls  attention  to  two  principles  that  aid  very  much  in 
"  staking  out  "  earthwork. 

The  logarithmic  tables  are  not  reproduced,  for  the  reason 
that  they  are  but  little  used  and  should  not  be  used  at  all — 
and  most  emphatically  so  in  this  line  of  work.  It  is  fair  to 
observe  that  the  space  required  for  such  tables  is  replaced  by 
numerous  useful  tables,  applying  directly  to  the  matter  in  hand, 
and  also  to  enlarging  the  subject-matter  of  the  book,  thus  sav- 
ing greatly  in  time  and  labor.  Furthermore,  it  should  be  stated 
that  there  is  no  problem  in  the  book  requiring  a  computation 
more  complex  than  to  find  the  cost  of  29  oranges  (say),  sup- 
posing that  17  oranges  cost  43  cents.  The  author  must  believe 
that  no  person — much  less  an  engineer — would  think  of  apply- 
ing logarithms  to  the  above  example;  and  if  so,  he  could  not 
with  any  propriety  apply  them  to  any  problem  in  the  Manual, 
since  the  nature  of  the  numerical  computation  to  be  made,  and 
not  the  subject-matter  of  the  problem,  furnishes  the  test  of 
methods. 

The  author  invites  criticism,  and,  should  another  edition  be 
called  for,  will  make  the  best  use  possible  of  any  suggestion 
that  may  in  good  faith  be  made  to  him. 

The  Tables,  all  but  three,  were  computed  expressly  for  this 
book,  and  scrupulous  care  has  been  taken,  by  numerous  readings 
and  checks,  to  make  all  tables  exact  to  the  last  figure. 

The  book,  as  indicated  in  a  few  places,  has  be"en  in  prepara- 


VI  PREFACE. 

tion  several  years  and  contains  matter  gathered  all  along  the 
paths  of  the  author's  experience;  and  is  the  result  of  a  belief, 
on  his  part,  of  his  ability  to  aid  his  professional  brethren  in  this 
direction. 

While  this  delay  has  not  been  to  the  advantage  of  the  author, 
it  lias  nevertheless  given  opportunity  for  due  reflection  and  re- 
consideration; and  therefore,  as  a  work  of  judgment  based  on 
experience,  the  volume  is  offered  to  his  brother  enquirers.  If 
the  book  even  partially  accomplishes  the  object  of  the  author's 
aims,  lie  will  feel  that  the  days  he  has  devoted  to  it,  though 
many,  have  not  been  spent  in  vain. 

P.  H.  PHILBKICK. 


CONTENTS. 


CHAPTER    I. 

PRELIMINARY  OPERATIONS. 

PAGE 

The    Reconnoissance 5 

The    Preliminary   Survey 5 

The  Location    5 

The   Organization   of  the  Transit    Party 8 

The  Compass:    What  kind  to  use  and  when  to  use  it 8 

Requirements  for  a  Successful  Reconnoissance 9 

Train    Resistances 10 

Total  Ascent  the  Main  Test  of  Gravity  Resistance n 

CHAPTER    II. 
ADJUSTMENTS,  USE,  AND  CARE  OF  INSTRUMENTS. 

The  Transit. 

Adjustments    12 

Use  and  Care  of  the  Transit 15 

Best    Way  to   Set  up   the   Instrument 15 

To  Measure  the  Angle  between  Two  Lines  or  Objects 16 

Hints  on  the  Care  of  Transits  and  Other  Instruments...                   .  16 


The  Level. 


Adjustments    

Use  of  the  Level. 


The  Compass. 

Adjustments    22 

Use   of   the    Compass 23 

vit 


Viii  CONTENTS. 

CHAPTER   III. 

PLANE  TRIGONOMETRY. 

PAGE 

Definitions   and   Explanations 25 

Fundamental    Relations 26 

Solution  of  Plane  Right  Triangles 31 

Table   for    Plane    Right   Triangles 32 

Fundamental    Relations   for    Oblique    Triangles 32 

Short  Solution  of  "  Tangent  Problem  " 34 

Solution  of  Plane  Oblique  Triangles 35 

Table  for  Solution  of  Oblique  Triangles 36 

List   of   Fundamental    Formulas 37 

CHAPTER    IV. 
SIMPLE  CURVES  CONNECTING  RIGHT  LINES. 

Properties  Relating  to  the  Circle 39 

Some  Elementary  Relations 41 

Notation 41 

Degree  of  Curve  Defined  and  Explained 41 

Rational  Treatment  of  Curves,  Example  Illustrating 42 

Difference  in  Lengths  of  Arcs  and  Subtended  Chords 43 

Huygens'  Formula  for  the  Length  of  an  Arc 44 

Table  Showing  Excess  of  Arcs  over  Subtended  Chords  when  the 

Arcs  are  Aliquot  Parts  of  100 44 

Table  Showing  Excess  of  Sub-chords  over  Aliquot  Parts  of  100, 

when  the  Chord  =  100 45 

Reason  for  these  Large  Excesses  Pointed  Out 46 

Formulas  for  Radius,  Tangent,  External  Secants,  Offsets,  etc 47 

Proper  Course  to  Pursue  in  Locating  a  Curve 49 

Long  Chords  and  Ordinates  to  Long  Chords 50 

Approximate  Value  of  Ordinates  to  Short  Chords 51 

Offsets  in  Terms  of  the  Degree  of  a  Curve 52 

Applications  of  Formula 53 

Laying  Out  Curves. 

A.  By  Deflection    Angles 53 

B.  By  Tangent   Offsets.     New   Method.     Without   Calculation 55 

C.  By  Ordinates   from   a   Long   Chord.     Without    Calculation 57 

D.  By  Chord  Offsets.     Without   Calculation 58 

E.  By  Middle    Ordinates.      Without    Calculation 59 

F.  By  Radial   Lines.     Without   Calculation 59 

Errors  in  Field-work — The  Nature  of. 

"A"   Method  of  Laying  Out 60 

"B"   Method   of  Laying  Out. 64 

"C"    Method   of   Laying   Out 64 

"  D  "   Method  of  Laying  Out 64 


CONTENTS.  IX 

PAGE 

"E"   Method  of  Laying  Out 65 

Fourteen    Problems    in    Simple    Curves - 65 

Obstacles  in  Surveying. 

To  Erect  a  Perpendicular  at  Any   Point  of  a   Line 76 

Table  and   Formulas  giving  Sides  of   Right   Triangles 77 

To  Drop  a  Perpendicular  from  a  Given  Point  to  a  Given  Line 77 

To  Draw  a  Perpendicular  to  a  Line  from  an  Inaccessible  Point....  77 

To  Prolong  a  Line  past  an  Obstacle  and  to  Measure  its  Length...  78 

Obstacles  to  Measuring  a  Line. 

When    One   End    is    Inaccessible 79 

When    Both    Ends   are    Inaccessible 80 

When   an   Inaccessible   Space   Intervenes 81 

Rest  Method  of  Making  a  Preliminary  Survey KJ 

Table    Illustrating    the    Same 83 

To    Replace    a    Broken    Line    between    Two    Points    by    a    Straight 

Line 85 

To  Find  the  Angle  between  Two  Straight  Lines   when  the   Point  of 

Intersection    is    Inaccessible 86 

To    Connect   Two   Tangents   by   a    Curve   when    the   Vertex   is    Inac- 
cessible   87 

To  Locate  a  Curve  when  the  Vertex  and  Both   Ends  of  the  Curve 

are    Inaccessible 87 

To    Pass    from    Any    Point    on    the    Curve    to    Any    Point    on    the 

Tangent    87 

To  Find  any  Desired  Point  on  the   Curve  when   Obstacles  preclude 

ti.e    Use    of    Ordinary    Methods 88 

CHAPTER    V. 
LEVELING,  STADIA  MEASUREMENTS,  ETC. 

Bench,    or    Bench-mark 90 

Form   of  Field-book  for  Level   Notes 92 

Proof  for   Level   Notes 93 

Benches:     Where    to    Establish    them 93 

The  Location  of  a  Level   Line 94 

The  Location  of  a  Grade  Line 94 

Correction  for   Curvature  and   Refraction 95 

Trigonometric    Leveling 97 

True  Simplified  Formulas  for  Heights  by  the  Dip  of  the  Horizon..  98 

The  Stadia. 

Formulas  for  the   Stadia — Simplified 102 

The  Gradient er. 

Formulas   for   the   Gradienter — Simplified 107 


Xll  CONTENTS. 


PAGE 

To  Connect  a  Straight  Track  and  a  Straight  Line  ...................   199 

Two  C'onnect  Two  Straight  Tracks  by  a  Curve  of  Three  I  tranches, 

the  Turnout  Curve  having  Radii  to  Suit  Given  Frogs  ............   199 

Formulas  Supposing  the  Switch-rail  to  be  a  Part  of  the  Turnout 

Curve    ................................................................  204 

A  Simple  Curve  Cannot  Meet  the  Conditions  Required  Above  ......  205 

Several   Methods  of  Laying  Out  Turnout  Explained  .................  207 

Double   Turnouts   from    a    Straight   Track  .............................  208 

To  Fit  a  Curve  to  a  Given  Middle  Frog  .............................  210 

Turnouts  from  Curves. 
Turnout  from   the   Inside  of  a   Curve  ..................................  212 

Turnout  from   the   Outside   of  a   Curve  ................................  213 

Double  Turnout  on   Opposite  Sides  of  a  Curve  .......................  214 

To   Find  Degree  of  a  Turnout  from  a  Curve  .........................  215 

Other   Turnouts   from   a   Straight   Track  ...............................  217 

To   Fit  the  Turnout  to  a  Given   Middle  Frog  ........................  219 

CHAPTER    IX. 
THE  TRUE  TRANSITION  CURVE. 


Reason  for  the  Need  of  Such   Curves 
Definition   and    Properties   of   Such    Curves 


Elementary  Relations. 
To    Find    the    Relative    Length    of    the    Offset    and    the    Transition 

Curve    ................................................................  225 

To    Find   Any   Tangent    Distance  ......................................  2_>6 

To    Find   Any   Offset  ....................................................  228 

Having  the  Offset,  t,  Supposing  the  Offset  Curve  of  the  Same  De- 

gree as  the  Original   Curve,   to   Find  the  True  Value,   /',   of  the 

Offset    ................................................................  230 

To  Find  the  Offset  iri  Terms  of  the  Cenrtal  Angle  and  the  Radius..  230 
To    Find   the   Angle   between    the   Tangent   and   Any    Chord    Drawn 

from    A  ...............................................................  23  1 

Cubic   Parabola  Not  Suitable  for  a  Transition   Curve  ................  232 

To    Find    Point    on    Curve    where    the    Tangent    is    Parallel    to    the 

Chord   of  the   Curve  .................................................  232 

To   Find  Tangents  at  the  Extremities  of  the  Curve  ..................  233 

To    Find    the    Length    of   Any    Radius    Vector,    or    Chord,    and    the 

Angles  between   these  Chords  ......................................  233 

To  Find  Deflection  Angle  at  Any  Point,  also  Any  Chord  ............  235 

To   Find  the   Exsec  dV  ',  also   TV,   etc  .................................  236 

To   Find  the   Radius  of  Curvature   at   Any   Point  .....................  236 

To  Lay  Out  the  Curve  by  Offsets  from  the  Tangent  AQ  ............  236 

Special  Problems  and  Examples. 

Given  Length  and  Degree  of  Kd,  to  Find  the  Offset  AK,  Tangent 
AO,  and  to   Lay  Out  the   Curve  ...................................  237 


CONTENTS.  xiii 


PAGE 

Given  the  Degree  of  the  Offset  Curve  and  Offset  AK,  to  Find  the 
Length  of  the  Transition  Curve,  etc 239 

Given  the  Degree  (Z)1)  of  Kd,  and  the  Tangent  AO,  to  Find  the 
Length  of  the  Transition  Curve  and  the  Offset  AK 239 

Given  the  Degree  of  the  Main  Curve,  and  the  Length  of  Kd,  to 
Find  the  Offset,  Tangent,  etc 241 

Given  the  Degree  of  the  Main  Curve  and  the  Length  of  Af  Re- 
placed, to  Find  the  Offset,  Tangent,  etc 242 

To  Replace  Each  Half  of  a  Simple  Curve  AfA1  by  a  Transition 
Curve  243 

To  Connect  Two  Tangents  by  Two  Transition  Curves,  Each  of  a 
Given  Length  5 244 

To  Connect  Two  Tangents  by  Two  Equal  Transition  Curves 
having  a  Common  Vertex  Distance  E 245 

The  Transition  Curve  Very  General  in   Use 245 

To  Lay  Out  the  Curve  from  Any  Point  on  it 245 

To  Substitute  a  Transition  Curve  for  Each  End  Portion  of  the 
Main  Curve  without  Changing  the  Rest  of  the  Curve 246 

The  True  Transition  Curve  Compared  with  Some  Others  and  His- 
torical Note 251 


CHAPTER    X. 

CALCULATION  OF  EARTHWORK. 

Prismoid   Defined 253 

Area  of  Level   Sections 254 

Area  of  Sections  not  Level 255 

Area  of  Irregular   Sections 256 

Formulas   for   Regular   Excavations   and    Embankments 256 

Error  of  the  "  End  Area  Volume  "  Always  Twice  the  Error  of  the 

"Middle  Area  Volume."     Demonstrated  by  Symbols 257 

The   End-area   Method    Simplified 258 

The    End-area  Volume    Generalized 259 

Special    Formulas   and    Cases 259' 

Formula    for    the    Volume    of    the    Frustum    of    a    Pyramid — Sim- 
plified      261 

Loaded  Flat  Cars,  Piles  of  Stone,  etc 263 

Ends  of  Embankments   or   "Dumps" 264 

Ground  Irregular  Laterally 264 

Mixed   Work,   Excavation,   and    Embankment 265 

Correction  of  Earthwork  for  Curvature 267 

Overhaul    268 

Monthly    Estimates 269 

Final    Estimates : 270 

Computation   of   Prismoids   Level   Laterally 270 


XIV  CONTENTS. 

CHAPTER    XT. 

APPROXIMATE  AND  ABRIDGED  COMPUTATIONS. 

PAGE 

Definitions    and    Notation 273 

The  Relative  Error 273 

Addition     274 

Subtraction    276 

Multiplication    and    Division 277 

Abridged    Multiplication 279 

Abridged    Division 281 

CHAPTER    XII. 

CONSTRUCTION. 

Clearing  and  Grubbing,   How  to   do  it 284 

Grade-line    284 

Surface   Ditches — Importance   of 285 

Cross-sections — Proper    Places   for 285 

Staking   Out   Earthwork   when   Ground   is   Level 286 

Staking  Out  Earthwork  when  Ground  is  Not   Level 287 

Two   Principles  to  Aid  in  Laying  Out   Earthwork 291 

Borrow-pits    - 291 

Shrinkage   of    Earthwork 294 

Retracing  the  Line 295 

Track-laying    296 

Culverts     297 

Location   of   Bridge   Piers 298 

Tunnels 298 

CHAFTEB    'Mil. 
EXPLANATION   OK  TARI.KS  AM.   MISCEI.LAKKOUS  TOPICS. 

To   Gauge  a   Stream  Approximately jo  i 

Transverse   Strength   of   Beams 302 

Safe    Bearing    Power   of    Piles 302 

TABLES. 

Table   for    Right    Triangles 32 

Table    for   Oblique    Triangles 36 

Table   Showing    Excess    of    Arcs   over    Subtended    Chords    when    the 

Arcs  are  Aliquot   Parts  of   100 44 

Table    Showing    Excess    of    Sub-cords    over    Aliquot    Parts    of    100 

when   the    Chord  =  100 45 

Table    Giving   the   Sides   of    Right    Triangles 77 

Table    for    Traverse    Survey 83 

Table  for  Level   Notes 92 

R.,  —  R, 
Table   Showing  the   Least  Value   of     T~-^—^-- 120 


CONTENTS. 


PAGE 

Table  Showing  Computation  of  Prismoids 271 

1.  Degrees,  Radii,  etc 304 

II.  Tangent   Offsets,   i   to   100   Feet 310 

III.  Offsets  for  Arcs  of  100  Feet 312 

Ilia.  Middle   Ordinate  Arcs  of   100  Feet 313 

Illb.  Chords  of  Arcs  of  100  Feet 313 

IV.  Long    Chords 314 

V.  Middle    Ordinates 316 

VI.  Turnouts  from  a   Straight  Track 318 

VII.  Tangents  and   Externals  of  a   i1   Curve 319 

VIII.  Arcs  of  Degrees,   Minutes,  and  Seconds  for  Radius  =  i 3.3 

IX.  Acres  for  Various   Lengths  and   Widths 323 

X.  Total    Grades 324 

XL  Correction   for    Curvature   and    Refraction 325 

XII.  Elevation   of  Outer  Rail 325 

XIII.  Coefficients  for  Stadia 3^6 

XIV.  Coefficients   for   Gradienter 327 

XV.  Offsets   for  Transition   Curves 328 

XVI.  Tangent    Distances    for    Transition    Curves 330 

XVII.  Deflection   Angles   for   Transition   Curves 332 

XVIII.  Earthwork  Tables,   Different   Slopes   and   Bases 333 

XIX.  Earthwork  Tables,  Two   Slopes  and  All   Bases 337 

XX.  Sines  and   Cosines 33'-$ 

XXL  Tangents  and   Cotangents 352 

XXII.  Versines  and   Exsecants 35<; 

XXIII.  Useful   Numbers  and  Formulas 382 

XXIV.  Conversion    of    Feet    into    Meters    and    Meters    into    Feet; 
also  Miles  into   Kilometers  and   Kilometers   into  Miles 383 


FIELD-MANUAL  FOE  ENGINEERS. 


CHAPTER  I. 

PRELIMINARY  OPERATIONS, 

1.  THE  engineering  operations  preparatory  to  the  construction 
of  a  railroad  are  : 

The  Reconnoissance  ; 

The  Preliminary  Survey  or  Surveys  ;  and 

The  Location. 

2.  The  Reconnoissance  is  a  general  but  incomplete  examination 
of  the  country  through  \vhich  the  proposed  road  is  to  pass,  made 
for  the  purpose  of  acquiring  data  upon  which  surveys  may  be 
made  and  compared,    and  the  best  possible  route  for  the  road 
selected. 

3.  A  Preliminary  Survey  consists  of  the  measurement  of  a  line, 
including  its  angular  deflections  ;  the  elevations  of  various  points 
upon  it,  the  determinations  of  the  topography  along  it  and  near 
it,  for  the  purpose  of  furnishing  the  data  from  which  the  line 
may  be  definitely  located;  or  the  survey  compared  with  other  sur- 
veys, for  the  purpose  of  selecting  one  from  which  the  location 
may  be  made. 

4.  The  Location  consists  in  placing  the  line  in  the  exact  posi- 
tion in  which  it  is  intended  to  be.     This  position  is  called  The 
Location. 

5.  It  is  convenient  to  carry  on  these  operations  concurrently. 
The  main  points  to  consider   in   the  location  are  the  relative 

cost  of  grading  and  bridging,  and  the  relative  grades  and  curva- 

K 


6  ITELD-MAKFAL  FOR  ENGINEERS, 

ture  of  the  lines.     In  grading,  the  character  of  the  soil  for  stabil- 
ity, in  both  "  cut "  and  "  fill,"  should  be  considered. 

It  is  sometimes  important  to  know  the  relative  value  of  property- 
traversed  by  different  lines  ;  and  if  the  lines  are  far  apart,  the 
probable  amount  of  traffic  that  the  respective  lines  can  command 
must  also  be  taken  into  account. 

6.  It  is  evident  that  the  best  possible  location  requires  the  least 
possible  grading,  bridging,  curvature,  etc.,  taken  together,  re- 
garding the   cost  and  the  expense  of  operating  the  road.     We 
can  afford,  therefore,  to  increase  the  curvature,  for  example,  if 
by  so  doing  we  can  at  the  same  time  decrease  the  earthwork,  and 
the  line  is  bettered  more  by  the  latter  than  it  is  damaged  by  the 
former. 

The  field-work  of  location  has  for  its  object  to  determine  the 
exact  position  of  the  selected  route  on  the  ground,  to  establish  the 
grade,  to  compute  the  amount  of  earthwork,  decide  upon  the 
amount  of  bridging,  etc. 

7.  A  railroad  line  usually  follows  the  valleys  of  watercourses 
or  the  dividing  ridges  between  watercourses,  or  crosses  valleys 
and  ridges  more  or  less  obliquely. 

8.  The  location  on  dividing  ridges  is  perhaps  the  simplest  of 
all.     In  this  no  bridges  and  few  culverts  are  required  ;  and  the 
elements  governing  the  location  are  mainly  the  amount  of  earth- 
work and  the  curvature  of  the  line.     In  this  case  a  sketch  of  the 
ridge,  especially  of  its  prominent  features  and  governing  points,  is 
made  while  walking  over  it;  and  the  preliminary  line  is  accurately 
run,  and  made  into  a  location,  if  the  route  is  adopted. 

9.  The  location  along  the  valley  of  a  stream  is  usually  more 
complex  than  the  former.     If  the  stream  is  so  small  that  the  cost 
of  bridging  it  would  be  plainly  less  than  the  advantage  to  the 
alignment  by  crossing  ;  or  if,  on  the  contrary,  the  river  is  so  large 
that  the  crossing  of  it  is  out  of  the  question,  the  -cost  of  bridging 
is  not  considered  and  the  problem  of  location  is  reduced,  in  the 
main,  to  that  of  making  the  best  alignment  within  the  limits  of 
the  valley  in  the  one  case,  or  upon  one  side  of  the  river  in  the  other. 
The  reconnoissance  and  surveys  would  be  made  as  already  de- 
scribed. 

Usually  the  most  favorable  ground  both  for  alignment  and  con- 
struction alternates  from  one  side  to  the  other  of  the  stream,  and 


PRELIMINARY  OPERATIONS.  7 

only  the  results  of  careful  and  scientific  surveys  can  tell  how 
many  and  where  the  crossings  rnust  be  in  order  to  make  the  align- 
ment and  grades  the  best  possible,  and  to  secure  the  most  favora- 
ble ground.  In  this  case  the  ground  on  both  sides  of  the  stream 
must  be  carefully  examined,  and  if  the  stream,  in  consequence  of 
banks  or  bottom,  is  not  easily  crossed  at  most  points,  the  most  suit- 
able crossings  must  be  found.  When  this  is  done  the  engineer 
will  mark  out  and  survey  one  or  more  lines  so  as  to  fit  the  chosen 
crossings  and  other  governing  points.  A  comparison  between 
different  lines  will  point  out  the  best,  and  generally  the  one  from 
which  the  best  location  can  be  made. 

10.  In  locating  a  line  across  valleys  and  ridges,  the  engineer 
must  find  the  best  crossings  of  the  streams,  and  the  gaps  or  notches 
in  the  ridges,  and  must  connect  such  of  the  former  with  such  of 
the  latter  as  will  furnish  the  best  line. 

11.  Sometimes  a  railroad  may  occupy  either  valleys  or  dividing 
ridges  for  the  greater  part  of  its  length  ;  in  which  case  a  choice 
must  be  made  between  the  higher  and  the  lower  line. 

The  higher  line  will  require  very  much  less  drainage  than  the 
lower  line,  which  is  an  important  advantage;  but,  on  the  contrary, 
probably  the  curvature  and  the  length  of  the  higher  line  will  ex- 
ceed that  of  the  lower. 

12.  It  is  not  to  be  supposed  that,  in  general,  a  railroad  will 
follow  either  a  valley  or  a  ridge;  or  that  it  will  cross  valleys  and 
ridges  obliquely  throughout  its   entire   length  ;   but  parts  of  the 
line  will  generally  do  so,  and  to  these,  and  therefore  to  the  whole 
line,  the  preceding  principles  will  apply. 

13.  The  regular  "  reaches  "  in  a  stream  furnish  the  best  cross- 
ings.    Throughout  these,  compared  with  other  points,  the  flow  of 
the  current  is  most  uniform;  the  wash  of  the  bottom,  and  the 
caving  and  the  shifting  of  the  banks,  are  least;  while  the  security 
of  the  foundations  of  bridges  and  of  approaches  is  greatest.     If 
ihe  bottom  of  the  stream  is  variable,  the  best  site  on  some  given 
"  reach  "  must  be  selected  with  a  view  to  the  kind  of  foundations 
f/.ui table  to  the  place.     Sharp  bends  in  streams  should  be  studi- 
ously avoided. 

.14.  The  engineer  should  freely  consult  the  best  maps  of  the 
(  (Kuitry  that  he  can  command,  and  he  should  prepare  a  map  on  a 


8  FIELD-MANUAL   FOR   ENGINEERS. 

convenient  scale,  upon  which  he  should  copy  the  principal  features 
of  the  country,  such  as  streams  and  lakes,  roads  and  towns,  and 
fill  in  the  details  as  he  progresses.  He  should  also  locate  on  the 
map  the  governing  points  of  the  route,  such  as  the  best  crossings 
of  streams,  the  "gaps"  in  the  ridges,  mountain  passes,  etc.  lie 
may  then  sketch  the  line  on  the  map. 

In  a  densely  wooded  country,  the  making  of  a  thorough  recon- 
noissance  is  comparatively  difficult.  In  this  case  it  will  often  be 
necessary  to  cross  and  recross  the  country  many  times  before  a 
comprehensive  knowledge  of  it  can  be  gained. 

15.  The  one  almost  indispensable  instrument  in  making  a  re- 
connoissance   is   a   pocket-compass.      Field-glasses,   hand-levels, 
telemeters,   and  other  instruments  are  sometimes  used,  but  are 
rarely  needed,   and   cannot  be  used  to  advantage.     Those  who 
cannot  make  a  proper  reconnoissance  without  them  would  better 
employ  some  one  who  can,  and  turn  their  attention  to  other  parts 
of  the  work. 

16.  For   preliminary  surveys   the  corps  of  engineers  may  be 
constituted  as  follows:  A  chief  engineer  or  engineer  in  charge,  an 
assistant  engineer  or  transitinan,  a  levelman,  a  rodman,  a  stake- 
man,  a  rear  flagman,  two  chainrnen,  and  one  or  more  axmen  accord- 
ing to  needs.     The  head  flag  should  be  carried  by  the  head  chain- 
man. 

17.  Since  a  survey  can  be  made  more  rapidly  with  the  compass 
than  with  the  transit,  the  compass  may  be  used  in  preliminary 
work  to  a  limited  extent.     Owing,  however,  to  the  inherent  in- 
accuracies of  the  compass,  a  line  run  with  it  is  generally  worthless, 
except  as  a  guide  to  a  transit  preliminary,  from  which  the  location 
may  be  made.     The  compass  is  therefore  of  very  little,  if  any,  use 
in  a  comparatively  level  country,  but  is  useful,  if  at  all,  for  run- 
ning the  first  preliminary  line  in  a  billy  region,  where  several  lines 
must  be  run.     The  compass  should  be  light,  should  be  mounted 
on  a  Jacob 's-staff,  and  should  have  a  narrow  slit  in  one  sight,  and 
a  fine  platinum  wire,  or  its  equivalent,  stretched  along  the  center 
of  a  wide  slit  in  the  other  sight.     A  self-reading  rod  is  best  for 
this  work  because  it  saves  much  time  over  the  sliding  rod;  and 
because,  too,  it  enables  the  levelman  to  do  his  own  reading.     The 
ax  for  driving  stakes  should  have  a  broad  head.    .  Stakes  shotild 
be  of  a  generous  length,  say  from  30  to  36  inches,  well  driven  into 


PRELIMINARY  OPERATIONS.  9 

the  ground,  and  projecting  above  the  grass  and  other  vegetation, 
so  that  the  line  may  be  easily  followed  or  recovered,  in  fields  or 
woods,  by  the  engineer  and  others.  Short  stakes  occasion  the  loss 
of  much  time,  and  those  shorter  than  about  30  inches  are  generally 
a  nuisance,  except  on  streets,  well-traveled  roads,  etc. 

18.  To  make  a  successful  reconnoissance  requires  a  good  eye 
for  distances,  elevations,  etc. ,  and  a  quick  and  clear  perception  of 
the  salient  features  of  the  country.     One  must  be  able  to  form  a 
mental  picture  or  image  of  the  country  along  the  proposed  line ; 
and  of  a  number  of  lines  crossing  and  recrossing  each  other  within 
the  limits  of  his  mental  map.     Thus  he  may  be  able  to  sketch  a 
proposed  line,  or  several  proposed  lines,  which  at  any  time  may  be 
tested.     The  comparison  of  lines,  to  test  their  relative  economy, 
is  a  question  of  science,  aided  by  mathematics. 

19.  It  is  plain  that  without  skill  in  making  the  reconnoissance, 
the  surveys  would  be  at  first  more  or  less  at  random,  and  the 
reaching  of  a  location  roundabout  and  expensive.     On  the  other 
hand,  the  greatest  skill  without  a  sufficient  knowledge  of  mathe- 
matics and  without  a  knowledge  of.  the  principles  of  engineering 
bearing  upon  the  question,  cannot  produce  the  best  location. 

A  happy  combination  of  the  qualities  necessary  to  the  successful 
locating  engineer  is  most  assuredly  found  in  comparatively  few 
individuals. 

The  two  main  obstacles  to  contend  with  in  building  a  railway 
are  grades  and  curves.  Since  both  affect  the  cost  of  construction 
and  the  expense  of  operating  the  road,  such  grades  and  curves  as 
will  render  the  total  cost  of  construction  and  operation  of  road  a 
minimum  are,  as  already  suggested,  the  best.  Since  the  resistances 
due  to  grades  as  well  as  to  curves  add  to  other  resistances  to  the 
movement  of  trains,  and  since  it  is  often  possible  to  lessen  grades 
at  the  expense  of  curvature,  and  vice  versa,  it  becomes  necessary 
to  briefly  consider  the  nature  of  the  resistances  that  a  moving  train 
encounters,  with  a  view  to  compensation. 

The  resistances  to  overcome  are,  first,  the  friction  of  the  moving 
parts  of  the  engine  and  train ;  the  friction  of  the  wheels  on  the 
rails  ;  impacts  and  oscillations,  and  the  resistance  of  the  air. 
These  resistances  vary  with  the  condition  of  the  rolling  machinery, 
the  road,  and  the  weather,  and  are  not  accurately  known.  Fric- 
tion is  nearly  independent  of  the  speed  of  the  train;  but  the  resist- 


10  I'lKI.iJ-MANUAL   FOR   ENGINEERS. 

ances  due  to  impact  increase  with  the  speed,  and  those  due  to  the 
atmosphere  increase  in  a  still  greater  ratio. 

The  sum  of  these  resistances  on  a  level  track  in  fairly  good  con- 
dition and  average  fair  weather  is,  according  to  Vose, 


in  which  r  is  the  resistance  in  pounds  per  ton,  and  v  is  the  velocity 
in.  miles  per  hour.  It  is  not  to  be  supposed  that  this  formula  is 
accurate,  though  perhaps  it  is  as  nearly  so  as  any;  and  probably 
the  resistances  are  no  greater  than  the  formula  indicates.  "With 
a  velocity  of  20  miles  per  hour  the  formula  gives  r  =  10.34  pounds 
per  ton  of  the  entire  weight.  Under  unfavorable  conditions,  as  a 
wet,  soft,  and  rough  track  and  high  wind,  the  resistance  might 
be  20  to  40  per  cent  more,  or  even  greater  still. 

The  second  resistance  is  due  to  grades  of  the  track.  The  force 
necessary  to  overcome  this  is  such  a  part  of  the  total  weight  of 
the  train  as  the  vertical  rise  of  the  grade  is  to  its  length.  Let 
r'  —  resistance  per  ton,  and  U  =  the  ascent  per  100  feet.  Then 

?•'  =  —  ;  •  X  2000  pounds  per  ton.     For  a  \%  grade  (52.8  feet  per 
1  UU 

mile)  h  =  1  and  r'  —  20  pounds  per  ton,  the  force  necessary  to 
overcome  the  load.  A  \%  grade  would  of  course  create  a  resist- 
ance of  10  pounds  per  ton,  or  approximately  the  same  as  the 
resistance  caused  by  friction,  impact,  resistance  of  the  atmos- 
phere, etc.,  at  a  speed  of  20  miles  per  hour,  as  stated  above. 

The  third  resistance  is  due  to  the  curves  of  the  track.  This  re- 
sistance is  not  accurately  known.  On  American  roads  with  American 
rolling  stock  it  is  probably  about  one  half  a  pound  per  ton  for  each 
degree  of  curvature.  Letting  ?•''  represent  the  resistance  per  ton 
per  station,  and  D  the  degree  of  curvature,  the  resistance  per  sta- 
tion is  r"  =  .5D.  We  observe  that  the  resistance  of  a  1°  curve 
is  but  the  ^  part  of  that  due  to  a  \%  grade,  or  equivalent  to 

K<)     Q 

-^-  =  1  .32  feet  per  mile.     The  resistance  of  a  5°  curve  is  equiva- 

lent to  6.6  feet  per  mile,  and  that  of  a  10°  curve  to  13.2  feet  per 
mile,  etc.  The  resistance  offered  to  a  train  in  moving  through  100 
feet  of  a  1°  curve  is  one-half  pound  per  ton  moved  100  feet,  which 

is  equivalent  to  the  total  load  lifted  -^  X  100  =  ^  =  .025  ft.; 
and  since  the  resistance  varies  as  the  product  of  the  degree  of 


PRELIMINARY  OPERATIONS.  11 

curvature  and  length  of  the  curve,  or  as  the  total  curvature, 
the  resistance  on  any  curve  for  each  degree  of  change  of  direc- 
tion is  equal  to  the  lifting  of  the  train  ^  of  a  foot.  Since  an  en- 
gine can  haul  over  a  road  only  what  it  can  haul  over  the  most 
unfavorable  part,  easy  curves  are  necessary  where  heavy  grades 
must  occur,  and  vice  versa  ;  the  object  being,  of  course,  to  keep  at 
all  points  the  sum  of  the  resistances  due  to  grades  and  curves  be- 
low the  allowable  maximum.  In  equating  for  grades  and  curves, 
however,  the  ruling  element  is  largely  the  total  ascent  and  not 
the  slope  merely,  as  is  too  often  considered  to  be  the  criterion. 
No  more  power  is  required  to  make  an  ascent  of  20  feet,  for  ex- 
ample, on  a  2/c  grade  1000  feet  long  than  is  required  on  a  \%  grade 
4000  feet  long. 

To  compare  grades  simply,  that  is,  the  rale  of  ascent  or  descent 
especially  on  short  grades,  as  is  usually  done,  is  meaningless  at 
least — it  is  absurd. 

On  light  grades  no  allowance  usually  need  be  made  for  curva- 
ture, since  the  momentum  of  the  train  would  carry  it  over  the 
ascent  even  without  aid  from  the  engine.  Grades  situated  near 
stations,  however,  or  other  points  where  a  train  must  stop  or 
"  slow  up  "  are  greater  obstacles  than  in  other  situations  to  the 
movement  of  trains,  and  upon  such  grades  an  allowance  for  cur- 
vature, as  above  shown,  should  be  made. 

Every  locating  engineer  is  presumed  to  make  himself  familiar 
with  the  principles  involved  in  locations  before  making  them; 
and  to  aid  in  this  the  reader  is  referred  to  Vose  and  others,  and 
especially  to  the  elaborate  treatise  of  Wellington  on  "Railway 
Location."  Wellington  gives  in  Table  118  the  "  Total  Energy, 
or  Potential  Lift,  in  Vertical  Feet  in  Trains  moving  at  Various 
Velocities,"  and  discusses  at  length  all  subjects  connected  with 
location. 


CHAPTER   II. 

ADJUSTMENTS,  USES,  AND  CARE  OF  INSTRUMENTS. 

THE  TRANSIT. 

The  following  are  the  usual  adjustments  of  the  transit : 

A.  To  Adjust  the  Levels  (that  is,  to  place  the  levels  in  a  plane 
perpendicular  to  the  upright  axis). — Set  up  the  instrument  upon 
its  tripod  as  nearly  level  as  may  be;  unclamp  the  plates  and  hring 
the  two  levels  above  and  on  a  line  with  the  two  pairs  of  leveling- 
screws.     Then  by  means  of  one  pair  of  screws  bring  the  bubble 
of  the  level  above  them  to  the  middle  of  the  opening.     Without 
moving  the  instrument  bring  the  other  bubble  to  the  middle  in 
the  same  way.     Since  in  moving  one  pair  of  screws  very  far  the 
other  pair  is  liable  to  become  cramped  and   the   corresponding 
bubble  somewhat  disturbed,  it  is  advisable  to  bring  the  bubbles 
in  succession  near  the  middle,  repeating  if  necessary,  and  ending 
by  bringing  them  exactly  to  the  middle. 

When  both  bubbles  are  in  place  turn  the  instrument  through 
about  180°;  if  the  bubbles  are  now  in  position  they  need  no  cor- 
rection ;  but  if  not,  turn  the  small  nuts  at  the  end  of  the  levels 
until  the  bubbles  are  moved  over  one  half  of  the  error.  Then 
bring  the  bubbles  again  to  the  middle  by  the  leveling-screws  and 
repeat  the  operation  just  described  until  the  bubbles  will  remain 
in  the  middle  during  a  complete  revolution  of  the  telescope.  This 
shows  the  adjustment  to  be  completed. 

B.  To  Set  the  Cross-wires  Vertical  and  Horizontal.—  Level  the 
instrument.     Move  the  telescope  upward  or  downward  and  note 
whether  the  vertical  wire  traverses  some  fixed  point  or  not.     If 
not,  loosen  the  four  cross- wire  screws  and,  by  the  pressure  of  the 
hands  on  their  head  outside  the  tube,  move  the  cross- wire  ring 
around,  what  seems  to  be  sufficient,  and  repeat  the  operation,  if 
necessary,  until  the  vertical  wire  will  traverse  some  fixed  point. 

The  cross-wire  screws  are  near  the  screws  of  the  centering-ring 

12 


ADJUSTMENTS,  USES,  AND    CARE   OF   INSTRUMENTS.     13 

of  the  eyepiece  (the  beads  of  which  are  usually  covered  by  an 
outside  ring),  but  between  them  and  the  axis  of  the  telescope. 

C.  To  Make  the  Line  of  Collimation  Perpendicular  to  tlie  Axis  of 
tlie  Wyes,  so  tliat  it  will  revolve  in  a  plane. — Set  up  the  transit 
at  some  point  0  near  the  middle  of  a  level  piece  of  ground 
and  level  it  carefully.  Let  AOB  be  a  straight  line,  AO  and 
BO  being  nearly  equal.  Direct  the  line  of  sight  to  A,  clamp 
the  instrument  and  revolve  the  telescope,  and  if  the  line  of 
collimation  is  not  perpendicular  to  the  axis,  the  line  of  sight 
will  determine  a  point  0,  say,  on  one  side  of  J3.  To  test  the  mat- 
ter, loosen  the  upper  clamp  and  turn  the  vernier  plate  almost  or 
quite  half-way  around,  so  that  the  line  of  sight  will  again  be  on 


FTG.  1. 

A,  and  clamp  the  plates.    Again  revolve  the  telescope  and  note  tbe 
point  D,  suppose,  thus  determined.     If  the  point  C  coincides  with 

B,  D  will  also  coincide  with  B  and  the  line  of  collimation  is  in 
adjustment.     If,  however,  C  is  on  one  side  of  B,  as  shown,  D  will 
be  equally  far  on  the  other  side,  showing  the  line  of  collimation 
out  of  adjustment. 

To  correct  the  error,  use  the  two  capstan-head  screws  on  the  side 
of  the  telescope,  to  move  the  ring  to  which  the  wires  are  fastened 
laterally,  and  with  it,  of  course,  the  intersection  of  the  wires. 

Having  moved  the  vertical  wire  until  by  estimation  one  fourth 
of  the  space  DC  is  passed  over,  return  to  A,  clamp  and  revolve 
the  telescope,  and  if  the  correction  has  been  carefully  made  the 
line  of  sight  will  now  cut  the  point  B.  It  will,  however,  generally 
be  necessary  to  repeat  the  operation,  the  adjustment  not  being 
perfected  at  first. 

Remember  that  the  eyepiece  inverts  the  position  of  the  wires, 
and  therefore  in  moving  the  ring  the  operator  must  proceed  so  as 
seemingly  to  increase  the  error. 

The  wires  may  now  be  brought  into  the  center  of  the  field  of 
view  by  moving  the  screws  of  the  centering-ring  of  the  eyepiece, 
which  are  slackened  and  tightened  in  pairs,  the  movement  being 
now  direct  until  the  wires  are  seen  in  their  proper  position. 


4  tftELft-MANUAL   FOR   ENGINEERS. 

It  is  proper  to  observe  that  the  position  of  the  line  of  collirna- 
tion  depends  solely  upon  that  of  the  objective,  so  that  the  eye- 
piece may  be  moved  in  any  direction,  or  replaced  by  another 
without  at  all  deranging  or  in  any  way  affecting  the  adjustment 
of  the  wires. 

I).  To  Adjust  the  Standards  to  the  Same  Height  so  that  the  Line, 
of  Collima'.ion  icill  Revrtve  in  a  Vertical  Plane. — Set  the  transit 
as  close  as  convenient  to  the  base  of  a  lofty  spire,  or  other  high 
object;  level  it  carefully  and  clamp  it,  and 
direct  the  telescope  to  the  top  of  the  spire  or 
other  elevated  and  well-defined  point,  as  A 
in  the  figure.  B  is  in  the  same  vertical  plane 
with  A  and  the  instrument,  that  is,  with  the 
line  of  sight. 

Turn  down  the  telescope  to  some  good  point 
on  the  ground,  either  found  or  marked,  as  C. 
Unclainp  the  plates  or  spindle,  revolve  the 
telescope  and  turn  it  half-way  around,  or  far 
enough  to  again  sight  to  the  high  point. 
Again  clamp  and  turn  down  the  telescope  to 
~Q  some  point  D  opposite  C. 

In  setting  C  the  left  standard  must  have 
been  too  high,  and  in  setting  D  the  same 
standard  (now  the  right  by  revolving  the  telescope)  shows  to  be 
equally  high,  the  errors  EG  and  BD  being  equal.  Correct  the 
error  by  raising  or  lowering  the  sliding-piece  at  one  end  of  the 
axis  by  means  of  the  screw  beneath  and  those  above,  so  that  when 
the  telescope  is  directed  to  A  and  lowered  it  will  cut  B  half-way 
between  C  and  D.  If  the  instrument  is  in  adjustment  it  will,  of 
course,  cut  B  instead  of  C  and  D  in  the  first  two  trials. 

E.  To  Adjust  the  Vertical  Circle. — Set  up  the  instrument  firmly, 
level  it  carefully,  bring  into  line  the  zeros  of  the  circle  and 
vernier,  and  with  the  telescope  find  some  well-defined  point,  from 
two  to  five  hundred  feet  distant,  which  is  cut  by  the  horizontal 
wire.  Turn  the  instrument  half-way  around,  revolve  the  tele- 
scope, fix  the  wire  upon  the  same  point  as  before,  and  observe  if 
the  zeros  are  again  in  line.  If  not,  loosen  the  capstan-head 
screws  which  fasten  the  vernier,  and  move  the  zero  of  the  vernier 
over  half  of  the  error.  Bring  the  zeros  again  into  coincidence 
and  proceed  as  before,  as  many  times  as  necessary,  until  the 
error  is  entirely  corrected. 


ADJUSTMENTS,  USES,  AND    CARE   OF   INSTRUMENTS.     15 

F.  To  Adjust  the  Level  on  Telescope.— First  level  carefully  and 
clamp  the  telescope  approximately  horizontal  by  the  eye. 

Then,  Laving  the  line  of  collhnation  previously  adjusted,  drive 
a  stake,  say  two  or  three  hundred  feet  away,  and  note  the  height 
cut  by  the  horizontal  wire  upon  a  staff  set  on  top  of  the  stake. 

Fix  another  stake  in  the  opposite  direction  and  at  the  same  dis- 
tance from  the  instrument,  and  without  disturbing  the  telescope 
turn  the  instrument  upon  its  spindle,  set  the  staff  upon  the  stake, 
and  drive  the  stake  into  the  ground  until  the  reading  is  the  same 
as  on  the  first  stake.  The  tops  of  the  two  stakes  are  equally  high 
however  much  the  telescope  may  be  out  of  level. 

Now  set  the  instrument  some  twenty  or  thirty  feet  from  one  of 
the  stakes  and  on  the  prolongation  of  the  line  joining  the  two. 
Level  the  instrument,  clamp  the  telescope  as  nearly  horizontal 
as  may  be,  and  note  the  readings  of  the  staff  on  the  two  stakes. 
If  they  agree,  the  telescope  is  level.  If  they  do  not  agree,  then 
with  the  tangent-screw  move  the  wire  over  fully  the  whole  error, 
as  shown  at  the  distant  stake;  repeat  the  operation  just  described 
a  <  many  times  as  may  be  necessary,  so  that  the  wire  will  give  the 
same  reading  at  both  stakes,  showing  that  the  telescope  is  truly 
horizontal.  Taking  care  not  to  disturb  the  position  of  the  tele- 
scope, bring  the  bubble  into  the  middle  by  the  little  leveling-nuts 
at  the  ends  of  the  tube,  which  will  render  the  adjustment  com- 
plete. 

The  above  are  all  the  adjustments  usually  required  at  the  hands 
ofthe  engineer  ;  for  any  others  see  the  Manual  of  W.  &  L.  E. 
Gurley. 

USE  AND  CAKE  OP  TRANSIT. 

The  instrument  should  be  set  up  firmJy  with  the  plates  nearly 
level,  and  thus  save  much  time  in  turning  the  leveling-screws, 
besides  the  worse  than  useless  wear  upon  them. 

To  do  this  :  Set  up  the  transit  approximately  over  the  desired 
point.  Hold  firmly  the  leg  of  the  tripod  on  the  left  with  the  left 
hand,  and  move  the  foot  of  the  leg  on  the  right  with  the  right 
hand  in  any  direction,  so  as  to  bring  the  lower  plate  of  the  tripod 
head  approximately  horizontal,  determined  by  placing  the  eyes  in 
the  plane  of  its  upper  surface  and  sighting  as  nearly  as  may  be 
to  the  distant  horizon.  It  will  do  no  harm  to  bend  the  body  the 
trifle  necessary  to  do  this.  This  sight  is  so  long  that,  according  to 


16  FIELD-MANUAL   FOR   ENGINEERS. 

the  principle  of  the  "  division  of  errors,"  the  plate  can  be  readily 
set  very  nearly  horizontal  in  almost  any  position  of  the  instru- 
ment. 

Now  place  the  left  hip  against  one  leg;  grasp  the  opposite  leg 
with  the  left  hand,  and  the  leg  on  the  right  with  the  right  hand; 
and  keeping  the  relative  position  of  the  legs  unaltered,  move  the 
instrument  laterally  over  the  desired  point  and  lower  it  to  the 
ground. 

Of  course  the  two  movements  of  tilting  and  sliding  may  be  done 
together  and  in  much  less  time  than  required  to  describe  the 
movement. 

To  MEASURE  THE  ANGLE  BETWEEN  Two  LINES  OR  OBJECTS. 

Level  the  instrument  carefully;  bring  the  zeros  of  the  verniers 
and  limb  together  by  means  of  the  upper  clamp  and  tangent-screw, 
and  direct  the  telescope  toward  one  of  the  objects  by  means  of 
the  lower  clamp  and  tangent-screw.  Upon  loosening  the  upper 
clamp  and  directing  the  telescope  toward  the  second  object,  the 
angle  desired  is  then  shown  upon  the  limb. 

Before  making  an  observation  with  the  telescope,  the  eyepiece 
should  be  moved  in  or  out  until  the  cross- wires  may  be  distinctly 
seen.  This  may  be  accomplished  with  the  greatest  precision  by 
directing  the  telescope  toward  some  white  object.  The  sky 
will  serve  very  well  for  this.  The  objective  is  then  adjusted  by 
moving  it  in  or  out  until  the  object  is  seen  clear  and  well  defined, 
and  the  cross- wires  appear  as  if  fastened  to  its  surface. 

Water  and  dust  are  very  destructive  to  instruments  ;  indeed, 
the  most  destructive  of  all  agents  is  dust.  An  instrument  may  be 
badly  bruised,  bent,  or  broken,  by  a  fall  or  otherwise,  and  yet  it 
may  be  repaired.  If,  however,  it  is  allowed  to  stand  on  its  tripod 
in  a  dusty  office — and  all  offices  are  dusty — its  life  will  be  short. 
The  author  has  known  of  instruments  so  choice  that  their  owners 
would  allow  no  one  to  handle  them  lest  they  might  become 
slightly  soiled,  or  some  other  mishap  befall  them;  and  yet  in  a 
few  months  they  were  ruined,  in  the  way  pointed  out.  A  minute 
quantity  of  dust  on  the  sockets  will  probably  cause  them  to  grind; 
and  more  of  it  will  increase  the  grinding  so  that  the  instrument 
will  soon  become  useless. 

Whenever  not  in  actual  use  the  eyepiece  should  be  covered  by 
the  lid,  and  the  object-glass  by  the  cap,  to  protect  them  from  dust, 
moisture,  rain,  etc, 


ADJUSTMENTS,  USES,  AND   CAKE   OF   INSTRUMENTS.    17 

"When  an  instrument  is  exposed  to  the  hot  rays  of  the  sun  for 
some  time  its  parts  are  subjected  to  very  unequal  expansion,  which 
throws  the  instrument  more  or  less  out  of  adjustment,  thus  to  some 
extent  vitiating  the  work  and  damaging  the  finer  parts  of  the  in- 
strument. To  prevent  this  it  should  be  shielded  by  an  umbrella  or 
screen  of  some  kind  supported  on  top  of  a  high  stake;  or,  if  nothing 
better  is  at  hand,  a  cloth  should  be  placed  over  the  telescope. 

When  an  instrument  is  in  the  office  or  in  transport  it  should 
be  clamped  and  placed  in  a  stable  position,  upon  proper  supports, 
in  a  tight  box,  well  cushioned  if  possible. 

If  the  box  cover  does  not  fit  upon  the  box  closely  it  can  be 
remedied  by  sticking  suitable  cloth  on  the  top  edge  of  the  box  or 
on  the  under  side  of  the  lid. 

When  handling  an  instrument  it  should  be  supported  by  plac- 
ing the  hand  under  the  lower  plate. 

Tangent  and  micrometer  screws  should  be  used  equally  on  all 
portions  of  their  length. 

Keep  the  tripod  legs  tight  enough  on  the  tripod  head  to 
secure  stability  by  tightening  the  nuts  on  the  bolts  when 
necessary.  Also  tighten  the  shoes  of  the  tripod  if  they  become 
loose.  Neglect  of  these  things  sometimes  prevents  an  engineer 
from  keeping  his  telescope  on  a  point,  and  causes  him  to  run 
a  zigzag  line  without  knowing  the  cause  of  it.  Secure  the 
instrument  well  to  the  tripod  head  before  using  it;  and  bring 
all  four  leveling-screws  to  a  bearing  and  cover  the  instrument 
with  an  oilcloth  hood  before  carrying  it.  Use  a  fine  camel- 
hair  brush  or  a  piece  of  old  and  soft  linen  to  clean  the  glasses 
of  the  telescope. 

Dust,  moisture,  perspiration,  etc.,  will  sometimes  cause  a 
film  to  form  on  the  lenses  of  a  telescope  which  may  greatly  im- 
pair the  sight  through  it. 

To  remove  the  film,  the  lenses,  after  being  carefully  brushed, 
should  be  gently  wiped  with  a  piece  of  chamois-skin  moistened 
with  alcohol,  and  the  lenses  must  be  wiped  dry  by  using  fresh 
portions  of  the  skin  on  separate  parts  of  the  lens. 

To  remove  dampness  in  the  main  tube  of  the  telescope,  take 
out  the  eyepiece,  cover  the  open  end  with  cloth,  and  leave  the 
instrument  in  a  dry  room  for  some  time. 

The  centers  of  an  instrument  should  aluavs  be  lubricated 
with  fine  watch-oil  only,  and  after  a  careful  cleaning.  First 
wipe  off  all  old  grit  and  oil  before  applying  fresh  oil, 


18  FIELD-MANUAL   FOR   ENGINEERS. 

If  dust  settles  on  the  cross-wires,  unscrew  the  eyepiece  and 
the  object-glass,  and  gently  blow  through  the  telescope  tube; 
cover  up  both  ends  ahd  wait  a  few  minutes  before  replacing 
the  eyepiece  and  object-glass. 

Be  sure  to  bring  the  object-glass  cell  to  a  firm  bearing  against 
its  shoulder,  and  then  examine  the  adjustment  of  the  lines  of 
collimation. 

To  clean  the  threads  of  a  leveling  or  tangent  screw  use  a 
stiff  tooth-brush  to  remove  the  dust,  then  apply  a  little  oil 
and  turn  the  screws  in  and  out  with  alternate  brushing  to 
remove  dust  and  oil,  until  it  moves  freely  and  smoothly.  Use 
such  a  brush  to  clean  the  object-slide.  Xo  screws  should  be 
strained  more  than  necessary  to  insure  a  firm  bearing,  and 
this  applies  with  special  force  to  the  cross-wire  screws. 

All  straining  of  such  screws  beyond  this  impairs  the  accuracy 
of  the  instrument  and  the  reliability  of  adjustment. 

These  remarks  in  reference  to  the  care  of  the  transit  apply 
substantially,  of  course,  to  all  instruments. 

If  obliged  to  work  with  an  instrument  of  faulty  graduation, 
it  is  best  to  read  each  angle  on  different  parts  of  the  circle, 
and  take  the  mean  value.  If  Are  take  the  mean  result  obtained 
by  both  verniers,  we  eliminate  the  errors  due  to  eccentricity 
of  the  vertical  axis,  and  also  reduce  the  errors  of  graduation. 

For  greater  accuracy  clamp  the  vernier-plate  to  zero  and  read 
the  angle  by  both  verniers.  Then  keep  the  vernier-plate 
clamped,  point  the  telescope  to  the  first  object  and  proceed  as 
before,  any  number  of  times.  Then  read  the  verniers,  adding 
360°  for  each  complete  revolution  which  has  been  made,  and 
divide  this  sum  by  the  number  of  times  the  angle  was  read. 
The  quotient  is  the  required  angle. 

The  cross-wires  can  be  illuminated  easily  by  placing  a  piece 
of  white  cardboard,  with  a  hole  through  it  for  the  line  of  sight, 
in  front  of  the  telescope,  and  in  an  oblique  positio'n,  so  as  to 
reflect  into  the  telescope  the  rays  of  a  lamp  or  of  a  lantern 
placed  back  of  the  object-glass  and  near  the  telescope. 

THE  LEVEL. 

The  principal  adjustments  of  the  level  consist  in  the  follow- 
ing: 

I,  Bringing  the  cross-wires  into  the  optical  axis  of  (he  lele- 


ADJUSTMENTS,  USES,  AND    CARE   OF   INSTRUMENTS.     ID 

scope  and  consequently  parallel  to  the  line  of  bearings  of  the 
wye  rings. 

%2.  Causing  the  line  of  bearings  to  be  parallel  to  the  plane 
of  the  level. 

3.  flaking  either  of  these  lines,  and  therefore  all  of  them, 
parallel  to  the  bar  and  consequently  perpendicular  to  the 
axis  of  the  instrument. 

1.  To  adjust   the   line   of  collimation,  that  is,   to  bring  the 
cross-wires  into  the  optical  axis,  so  that  their  point  of  inter- 
section will  remain  on  any  given  point  during  an  entire  revolu- 
tion of  the  telescope. 

Set  the  tripod  firmly,  remove  the  wye-pins  from  the  clips, 
so  as  to  allow  the  telescope  to  turn  freely;  clamp  the  instru- 
ment to  the  leveling-head,  and  by  the  leveling  and  tangent 
screws  bring  cither  of  the  wires  upon  the  clearly  defined  edge 
of  some  object.  Then  carefully  rotate  the  telescope  half-way 
around. 

!  f  the  wire  docs  not  coincide  with  the  line  observed,  bring 
it  half-way  back  by  means  of  the  capstan-head  screws  at  right 
angles  with  it.  always  remembering  the  inverting  property 
of  the  eyepiece.  Then  by  the  leveling  and  tangent  screws 
bring  it  again  upon  the  "edge,"  and  repeat  the  above  opera- 
tion if  necessary,  and  continue  to  do  so  until  the  telescope  may 
be  rotated  without  changing- the  position  of  the  wires. 

If  both  wires  are  much  out  of  position,  it  will  be  well  to 
approximately  adjust  the  wires  alternately  before  attempting 
to  make  the  adjustment  of  either  one  complete,  since  an  error 
in  one  somewhat  affects  the  other. 

It  may  be  advisable  to  center  the  eyepiece.  To  do  so  unscrew 
the  covering  of  the  eyepiece  centering-screws,  and  move  each 
pair  in  succession,  with  a  screw-driver,  until  the  wires  are 
brought  into  the  center  of  the  field  of  view. 

The  inverting  property  of  the  eyepiece  does  not  affect  this 
operation,  and  the  screws  are  moved  directly.  To  test  the  cen- 
tering, rotate  the  telescope,  and  if  an  object  observed  appears 
to  change  position  the  centering  is  not  perfect. 

In  all  telescopes  the  line  of  collimation  is  determined  by  the 
cross- wires  and  objective,  and  is  not  affected  in  any  way  by  the 
eyepiece. 

2.  To  make  the  line  of  bearings  parallel  to  the  bubble-tube, 


20  FIELD-MANUAL   FOR   ENGINEERS. 

so  as  to  insure  that  it  is  horizontal,  when  the  bubble  is  in  the 
center.    This  adjustment* embraces  two  parts: 

'  First,  to  bring  the  center-line  of  the  bubble  and  the  line  of 
bearings  in  the  same  plane. 

Second,  to  make  these  lines  parallel. 

To  effect  the  first:  Clamp  the  level  and  bring  the  bubble 
to  the  center  by  the  parallel-plate  screw*.  Now  rotate  the 
telescope  in  the  wyes  20°,  more  or  less.  If  the  bubble  runs 
toward  the  end,  it  shows  that  the  center-line  of  the  bubble  and 
the  axis  of  the  telescope  are  not  in  the  same  plane;  in  other 
words,  the  bubble-tube  lies  crosswise  of  the  telescope. 

To  correct  the  error,  bring  the  bubble  by  estimation  half- 
way back,  by  the  capstan-'head  screws,  which  are  set  in  either 
side  of  the  level-holder. 

Again  bring  the  level-tube  under  the  telescope,  the  bubble 
to  the  middle,  etc.,  repeating  the  operation  just  described  until 
the  bubble  will  keep  its  position,  when  the  telescope  is  re- 
volved. 

For  the  second  part:  Bring  the  bubble  to  the  middle  of  the 
tube  by  the  leveling-screws,  and  take  the  telescope  out  of  the 
wyes  carefully  and  turn  it  end  for  end. 

If  the  bubble  runs  toward  either  end,  lower  that  end,  or 
raise  the  other  by  turning  the  adjusting-nuts  on  one  end  of  the 
level  until  by  estimation  half  the  correction  is  made.  Again 
bring  the  bubble  to  the  middle  by  the  leveling-screws,  and 
repeat  the  whole  operation  just  described  until  the  reversion 
can  be  made  without  causing  tiny  change  in  the  bubble. 

3.  Having  made  the  previous  adjustments,  it  remains  to 
make  the  level-bubble  (and  therefore  the  line  of  bearings  and 
line  of  collimation)  parallel  to  the  bar,  and  therefore  perpen- 
dicular to  the  vertical  axis  of  the  level,  so  that  the  bubble  will 
remain  in  the  middle  during  an  entire  revolution  of  the  tele- 
scope. 

Place  the  level  over  a  pair  of  leveling-screws,  and  by  means 
of  them  bring  the  bubble  to  the  middle.  Turn  the  instrument 
half-way  around  horizontally.  If  the  bubble  runs  toward  either 
end,  bring  it  half-way  buck  by  either  pair  of  nuts,  at  the  ends 
of  the  bar.  Then  bring  the  bubble  to  the  middle  again,  by  the 
level  ing-screws,  etc..  repeating  the  operation  just,  described, 


ADJUSTMENTS,  USES,  AND   CARE   OF   INSTRUMENTS.     21 

until  the  bubble  will  remain  in  the  middle  of  the  tube  when 
the  instrument  is  revolved. 

In  making  this  adjustment  it  is  best  to  use  the  opposite 
pairs  of  leveling-screws  alternately,  thus  bringing  the  upper 
parallel  plate  of  the  tripod  head  into  a  position  as  nearly  hori- 
zontal as  possible,  so  that  the  error  caused  by  not  revolving  the 
instrument  precisely  180°  may  be  the  least  possible. 

This  adjustment  is  for  convenience  and  not  for  accuracy  in 
any  appreciable  degree. 

Xow  turn  the  telescope  in  the  wyes  until  the  pin  on  the 
clip  of  the  wye  will  rest  in  the  little  recess  in  the  ring  to  which 
it  is  fitted. 

Apply  the  horizontal  wire  to  any  level  line,  and  in  case  it 
does  not  coincide  with  it,  loosen  two  cross-wire  screws  at  right 
angles  to  each  other,  and  by  their  heads  outside  turn  the  cross- 
wire  ring  until  the  horizontal  wire  coincides  with  the  level 
line. 

The  line  of  collimation  must  then  be  adjusted  again.  In 
readjusting  the  line  of  collimation  none  of  the  lines  referred  to 
in  making  the  adjustment  is  disturbed,  and  the  adjustments 
are  complete. 

ADJUSTING  BY  THE  "  PEG  "  METHOD. 

By  this  method  the  main  adjustments  are  effected  at  once. 

Drive  two  pegs  several  hundred  feet  apart,  and  set  the  in- 
strument midway  between  them.  Read  the  rod  on  each,  keep- 
ing the  bubble  in  exactly  the  same  position,  preferably  at  the 
center.  The  difference  of  the  readings  is  the  difference  of  the 
heights  of  the  pegs,  no  matter  how  much  or  in  what  way  the 
level  may  be  out  of  adjustment.  Then  set  over  either  peg  and 
measure  the  height  of  cross-wires  above  top  of  peg. 

The  difference  of  heights  of  pegs,  added  to  this,  or  taken 
from  it,  according  as  the  instrument  is  over  the  higher  or  lower 
peg,  gives  the  height  of  the  cross-wires  above  the  other  peg. 
Set  the  rod  on  that  peg,  and  bring  the  horizontal  wire  to  that 
height  on  the  rod  by  the  leveling-screws,  keeping  them  at  a 
bearing.  Then  bring  the  bubble  to  the  center  by  raising  or 
lowering  one  end  of  the  level-tube. 

The  first  part  of  the  second  adjustment, — namely,  to  bring 


22  FIELD-MANUAL   FOR   ENGINEERS. 

the  level-bubble  and  line  of  collimation  in  the  same  plane, — 
also  the  third  adjustment,  should  be  made  as  heretofore. 

USE  OF  THE  LEVEL. 

The  instrument  should  be  set  up  firmly,  with  the  top  of  the 
tripod  as  nearly  level  as  may  be,  so  as  to  save  time  in  leveling 
and  the  wear  of  the  screws,  etc. 

The  setting  up  of  the  level  is,  of  course,  similar  to  that  of  the 
transit  already  described. 

The  bubble  should  then  be  brought  over  each  pair  of  level- 
ing-screws  successively,  and  leveled  in  each  position. 

Bring  the  wire  precisely  in  focus  by  the  eyepiece,  and  the 
object  distinctly  in  view  by  the  objective,  so  as  to  avoid  all 
"  traveling  of  the  wires  "  or  parallax. 

It  is  best,  where  practicable,  to  take  approximately  equal 
fore  and  back  sights,  so  as  to  eliminate  any  error  due  to  a  lack 
of  perfect  adjustment,  Athich  is  difficult  to  secure. 

For  precise  reading  the  rod  should  not  be  over  400  or  500 
feet  from  the  instrument. 

If  the  socket  of  the  instrument  sticks  in  the  leveling-head  so 
as  to  be  difficult  to  remove,  be  sure  that  the  instrument  is  un- 
clamped  and  the  leveling-screws  are  free.  Then  place  the  palms 
of  the  hands  under  the  wye-nuts  under  each  end  of  the  bar, 
and  give  a  sudden  upward  blow  to  the  bar,  and  take  care  also 
to  grasp  it  the  moment  it  is  free. 

To  ADJUST  THE  COMPASS. 

The  Levels. — First  bring  the  level-bubbles  into  the  middle 
by  the  pressure  of  the  hands  on  different  parts  of  the  plate: 
then  turn  the  compass  half-way  around.  If  either  bubble  runs 
toward  one  end  of  its  tube,  it  indicates  that  that  end  is  too 

high- 
Lower  it  by  loosening  the  screw  under  the  lower  end,  and 
tightening  the  one  under  the  higher  end,  until  the  error  is, 
by  estimation,  half  removed.  Level  the  plate  again,  and  repeat 
the  operation  until  the  bubbles  will  remain  in  the  middle  dur- 
ing an  entire  revolution  of  the  compass. 

The  Sight-ranes.— The  sights  may  next  be  tested  by  observ- 
ing through  the  slits  a  fine  hair  or  thread  made  exactly  verti- 
cal by  a  plummet. 


ADJUSTMENTS,  USES,  AND    CAKE    OF    INSTRUMENTS.     23 


If  either  slit  does  not  coincide  in  direction  with  the  hair  or 
thread,  it  must  be  made  to  do  so  by  filing  its  under  surface  on 
the  higher  side. 

The  Needle. — Having  the  eye  nearly  in  the  same  plane  with 
the  graduated  rim  of  the  compass  circle,  observe  whether  or  not 
the  ends  of  the  needle  cut  opposite  points  on  the  rim.  If  not, 
bend  the  center-pin  (by  means  of  a  small  wrench)  about  an 
eighth  of  an  inch  below  the  point  of  the  pin,  so  as  to  make  the 
ends  cut  opposite  points  on  the  rim. 

The  needle  now  may  be  supposed  to  occupy  the  position 
Npti,  the  pivot  p  not  being 
in  the  center  0.  Now 
keeping  the  needle  in  the 
same  position,  turn  the 
compass  half-way  around. 
The  needle  will  now  oc- 
cupy the  position  N'p'S'. 
Correct  half  the  error  by  El 
bending  (that  is,  straigJit- 
CH'UH/)  the  needle,  making 
it  cut  points  half-way  be- 
tween its  former  positions, 
so  that  it  occupies  the  posi- 
tion Ap'B  and  is  straight; 
and  correct  the  other  half 
by  bending  the  pin,  placing 

the  point  of  the  pin  at  0  and  giving  the  needle  the  position 
NOS. 

The  operation  should  be  repeated  until  perfect  reversion  is 
secured  in  the  first  position. 

Then  try  the  needle  on  another  quarter  of  the  circle,  and  if 
an  error  is  manifested,  correct  the  center-pin  only,  the  needle 
being  already  straightened  by  the  previous  operation. 

Do  the  same  on  other  quarters  of  the  circle  until  the  needle 
will  reverse  in  any  position. 

To  USE  THE  COMPASS. 

In  using  the  compass  keep  the  south  end  toward  the  person, 
and  read  the  bearings  from  the  north  end  of  the  needle. 
Mark  every  station  or  point  at  which  the  compass  is  set,  so 


24  FIELD-MANUAL   FOll   ENGINEERS. 

that  it  may  be  easily  found  for  verification  or  a  resurvey.  It  is 
much  more  important  to  have  the  compass  level  laterally,  or 
crosswise  of  the  sights,  than  in  their  direction;  since  if  it  is 
not  so,  on  looking  up  or  down  hill  through  the  lower  part  of 
one  sight  and  the  upper  part  of  the  other  the  line  of  sight  will 
not  be  parallel  to  the  N.  and  S.  or  zero  line  on  the  compass, 
and  an  incorrect  bearing  will  be  obtained. 

A  continuous  line  thus  run,  to  say  nothing  of  other  im- 
perfections of  the  compass,  would  be  a  zigzag  line  probably 
very  much  in  error. 

The  compass  cannot  be  leveled  by  the  needle,  for  the  dip  of 
the  needle  is  continually  varying.  If  the  needle  touches  the 
glass  when  the  compass  is  leveled,  balance  it  by  sliding  the  coil 
of  wire  along  it. 

The  vibrations  of  the  needle  may  be  checked  by  gently  rais- 
ing it  off  the  pivot,  so  as  to  touch  the  glass,  and  letting  it 
down  again,  by  the  screw  on  the  under  side  of  the  box. 

The  compass  should  be  smartly  tapped  after  the  needle  has 
settled,  to  destroy  the  effect  of  any  adhesion  to  the  pivot  or 
friction  of  dust  upon  it. 

The  glass  sometimes  becomes  charged  with  electricity  by 
carrying  it  against  clothing,  or  wiping  it,  etc.,  so  that  it  at- 
tracts the  needle  to  its  under  surface,  preventing  its  free  move- 
ment. 

The  difficulty  may  be  remedied  by  breathing  on  the  glass, 
or  touching  it  in  different  places  with  the  moistened  finger. 

Of  course  the  chain  and  all  other  metals  should  be  kept  away 
from  the  needle. 


CHAPTER  III. 


PLANE  TRIGONOMETRY. 


1.  Plane  Trigonometry  treats  of  the  relations  of  the  sides  and 
angles,  and  of  the  solution  of  plane  triangles. 

2.  Some  French  writers  divide  the  right  angle  into  100  degrees, 
the  degree  into  100  minutes,  the  minute  into  100  seconds,  etc. 
This  centesimal  system,  in  which  reductions  'are  made  by  simply 
moving  the  decimal  point,  is  altogether  preferable  to  our  sexagesi- 
mal system. 

3.  Two  angles  whose  sum  is  equal  to  90°  are  complementary. 
Two  angles  whose  sum  is  180°  are  supplementary. 

4.  Let  us  consider  a  series  of  right  triangles  ABC,  AB'G',  etc., 
having  the  common  angle  A.     The  triangles  are  equiangular  and 
therefore  similar,  and  we  have 

BC      BG'      B"G" 


AB  ~  AB' 


AB" 


BG 
AG 


B'C'      B"C' 


AG 


AB  _  AB' 

AC  ~  AC' 


AC' 


AB" 


FIG. 


Thus  it  appears  that  the  ratios  of  the  sides  are  the  same  in  all 
right  triangles  having  the  same  acute  angles;  and  therefore  if 
these  ratios  are  known  in  any  one  of  these  triangles,  they  will  be 
known  in  all  of  them. 

As  any  triangle  may  be  divided  into  two  right  triangles,  it  is 
evident  that  the  solution  of  oblique  triangles  may  be  made  to  de- 
pend upon  the  solution  of  right  triangles. 

The  above  ratios,  depending  upon  the  angle  alone  and  not  at  all 
upon  the  absolute  lengths  of  the  sides,  may  be  considered  as  indices 

25 


2G  FIELD-MANUAL   FOR  ENGINEERS. 

of  the  angle,  and  have  received  special  names,  which  we  will  ex- 
plain. 

5.  Let  us  represent  the  sides  and  angles  of  a  triangle  in  the 
usual  way,  shown  in  Fig.  5.  The 
side  opposite  an  angle,  divided  by 
the  hypothenuse,  is  called  the  sine  of 
that  angle.  Thus 


—  =  sin  A, 
c 


and     -  =  sin  B. 
c 


The  side  opposite  an  angle,  divided  by  the  adjacent  side,  is  called 
the  tangent  of  that  angle.     Thus 


The  hypothenuse,  divided  by  a  side  adjacent  to  an  angle,  is 
called  the  secant  of  that  angle.     Thus 


v  . 

—  =  sec  A; 
b 


-  =  sec  3. 
a 


6.  The  terms  cosine,  cotangent,  and  cosecant  are  convenient 
abbreviations  for  the  "sine  of  the  complement,"  "tangent  of  the 
complement,"  and  "  secant  of  the  complement, "  respectively.  The 
reader  must  not  suppose  that  there  is  such  a  thing  or  entity  as  co- 
sine, cotangent,  or  cosecant  of  an  angle. 

Since  the  acute  angles  of  a  right-angled  triangle  are  comple- 
mentary, that  is,  A  =  90°  —  B  and  B  =  90°  —  A,  it  follows  that 

cos  A  means  the  sine  of     (90°  —  A),  or  sin  B; 
cot  A  means  the  tangent  of  (90°  —  A),  or  tan  B; 
cosec  A  means  the  secant  of  (90°  —  A),  or  sec  B,  etc. 


Hence 

sin  J.  = 


cos  5=; 

c 


tan  A  —      cot  B  =•  —  > 


cos  A  =  sin  B  =  — ; 


cot  A  =  tan  B  =  — ; 
a 


(4) 


sec  A  —  cosec  B  =  r ;    cosec  A  =  sec  B  =  - . 
h  a 


PLANE   TRIGONOMETRY. 


Sin  ,4  =  cos  B  is  really  an  identical  equation,  since  cos  B  is  the 
sine  (90°—  B)  =  sin  A;  and  so  is  tan  A  =  cotB,  etc.  Furthermore, 

c  —  b 
vers  A  =  =  1  —  cos  ^L. 


exsec  A  =  - — ; —  =  sec  A  —  1  = 


cos  A 


1  —  cos  A      vers  A 


cos  J. 


cos  A' 


B 


7.  Plane  trigonometry,  applied  to  plane  triangles,  is  but  the  ap- 
plication of  the  one  well-known  proposition  in  geometry:  "  Equi- 
angular triangles  have  their  homologous  sides 

proportional  and  are  similar"." 

Comparing  Figs.   5  and  6,  we   observe  that 

the  above  ratios  -,  -,  etc.,  change  as  the  angles 

A  and  B  change.  These  ratios  have  been  com- 
puted, however,  for  all  values  of  the  angles, 
differing  by  single  minutes  or  less,  and  placed 
in  tables  under  the  corresponding  headings, 
sine,  tangent,  etc. ,  and  opposite  the  correspond- 
ing  angles. 

8.  From  the  above  equations  we  observe  that 


b 
FIG.  6. 


Hence 


(6) 
(7) 


sin  A  cosec  A  =  sec  A  cos  A  =  tan  A  cot  A  =  1.     ,    (8) 
We  also  have,  by  division, 

sin  A       a 


cos  A       b 


3  FIELD-MANUAL  FOR  ENGINEERS. 

Again, 

sin' 4  +  cos' ;t=^+i' =<;=!.       .     .     .     (11) 
Again, 


6»  62 

and 


sec  A  =  4/1  -f  tan2  A ; (12) 

cosec2  A  =  sec2  jB  =  —  = —  =  1  -f-  —  =  !-}-  cot2  ^4, 

and 

cosec  A  = 
From  (9), 

tan 
sin  A  =  cos  A  tan  J.  = 


From  (10), 

cos  J.  =  sin  J.  cot  J.  — 


cosec  . 


If  one  of  the  above  functions  of  the  angle  A  is  given,  all  the 
others  may  easily  be  found.  For  example,  if  sin  A  is  given,  we 
have,  from  (11), 


cos  A  =  |/1  —  sin5  A. 
Then 


sm  A  sin  A 

tan  J.  = T=  -  =  J  •     .     =     •     (16) 

cos  J.         /i  _  sins  j. 


sm  A  sin 


——  =  -—  :          —        ... 
cos  A       4/1  _  sin2  A 


(18) 


PLANE   TRIGONOMETRY. 


If  cos  A  is  given,  we  have,  from  (11), 


sin  A  —  4/1  —  cos*  A. 

Then  tan  A,  etc.,  as  above. 

Similarly  when  other  functions  of  the  angle  are  given. 

9.  The  sine  and  cosine  of  two  angles  being  given,  to  find  the 
sine  and  cosine   of  their  sum,    and  the 
sine  and  cosine  of  the  difference  of  their 
angles. 

In  Figs.  7  and  8  let  EOF  =  A,  and 
FOE=B;  then,  in  Fig.  7,  EOG=A+B, 
and,  in  Fig.  8,  EOO  =  A  -  B.  From 
any  point  F  in  OF  draw  FE  perpen- 
dicular to  OF;  also  EG  and  FH  per- 
pendicular to  OH,  and  FK  perpendicular 
to  EG.  Now  the  three  sides  of  the  tri- 
angle FEK  are  perpendicular  to  the  three  sides  of  the  triangle 
FOH,  and  therefore  FEK  =  FOH. 

No\v,  in  Fig.  7, 


FIG. 


!„  (A 


EK 


EO 


But 


and 


FH      FH   FO 


Ji/Ji 


=  cos  A  sin  B. 


.  '.     sin  (  A  +  B)  =  sin  A  cos  B  -f-  cos  A  sin  B,    .     . 
Again, 

.       OG      OH-KF     OH      KF 


.     (20) 


OR    OF       KF   EF 


=  C°S  A  Cos  3  ~  sin  A 


30  FIELD-MAKUAL   FOR  ENGINEERS. 

Then,  in  Fig.  8, 

_  EG  _  HF-KE_  HF     OF  _  KE    FE 
sm(A-B)=~gQ-        EQ       ~  OF' ^EQ~  ~YE'~EO 


=  sin  A  cos  B  —  cos  A  sin  B\ 


(22) 


OG 
cos  (A-B}  =         = 


FK_OH    OF^       FK^    FE 

1      ~   OF'  E0~^~  FE  '  OE 


—  cos  A  cos  B  +  sin  A  sin  B. 


(23) 


0  H          G 

FIG.  8. 

In  (20),  make  B  =  A  and  get 

sin  2A  =  2  sin  A  cos  A.  (24) 
In  (21),  make  B  =  A  and  get 
cos  2  A  =  cos2  A  —  sin2  A 

=  (1  —  sin2  A)  —  sin9  A  —  1  —  2  sin8  A. 
=  cos2  J.  -  (1  -  cos2  A)  =  2  cos2  ^.  -  1. 

(20)  and  (21)  give 

sin  A         sin  B 

sin  (J.  -j-  B)  _  sin  .4  cos  7? -f  cos  AsinB_      cos  A         cos  B 
cos  (A  +  B)  ~  cos  .4  cos  B  —  sin  .A  sin  B 


1  - 


cos 


X 


sin  B  ' 


or 


tan  ( A  -f  B)  = 


tnn 


tan  B 


I  —  tan  A  tan 


Similarly,  from  (22)  and  (23), 


tan  (A  ~  B)  =  ~ 


tan  A  —  tan  B 


1  -|-  tan  A  tan  B 


(27) 


(28) 


PLANE   TRIGONOMETRY.  31 

We  note  some  special  values  of  the  trigonometric  functions. 
See  Fig.  9. 

Let  COP  represent  any  triangle  ;  angle  COP  =  0.  Let  0  =  0. 
Then  CP  =  0,  PO  and  BO  coincide  and  are  equal. 

Hence 

SinO  =  ^  =  0,      tanO  =  JL  =  0,      «c  0  =  ™  =  I. 

Let  0  =  90°.    Then  PC  and  PO  coincide  with  B'O,  and  CO  =  0. 
Hence 

~£)f  f\  T>  ' /")  T>/  /^) 

sin  90°  =  —r  =1,     tan  90°  =  — —  —  oo ,     sec  90°  =  ~-  =  oo . 

_O    U  0 

Let  0  =  180°.    Then  PO  and  CO  coincide  with  B"0  and  PC  =  0. 
Then 

sin  180°  =^L  =  0,     tan  180°  =  ^  =  0,     sec  180°=  |^  =  1. 

Let  0  =  270°.  Then  PO  and  PC  coincide  with  B'"0  and  CO  =  0. 
Hence 

7?"'O  7?//7O  Ti"fO 

sin  270°  =  57^=1,     tan  270°=  —  -- =00  ,     sec  270°  =^r-^=  oo  . 
1>     U  0 

The  values  of  the  functions  of  360°  are  the  same  as  those  of  0°. 
We  need  not  consider  angles  greater  than  180°. 

SOLUTION  OF  PLANE  RIGHT  TRIANGLES. 

16.  In  order  to  solve  a  plane  right  triangle  it  is  only  necessary 
to  select  from  equations  (4)  an  equation  containing  the  two  given 
parts  aside  from  the  right  angle  and  the  part  sought.  By  trans- 
posing, if  necessary,  so  as  to  express  the  latter  in  terms  of  the 
former,  it  becomes  known.  There  are  two  cases  : 

CASE  I. — A  side  and  an  angle  given.  Given  A  and  c.  See 
Fig.  5. 

Example.— Let  A  =  35°  23',  and  c  =  874.8.     We  have 

B  =  54°  37'. 
Also  table  of  sines  and  cosines  gives 

sin  35°  23'  =  .57904,     and    cos  35°  23'  =  .81530, 

Hence  a  =  c  sin  A  =  874.8  x  .57904  =  506.54; 

b  =  c  cos  A  -  874.8  X  .81530  =  713.22. 


FIELD-MANUAL  FOR  ENGINEERS. 


CASE  II.— Given  two  sides. 

Example.— -Let  a  =  184.3,  and  c  =  246.     We  have 


sin  J.  =  _=  .74919. 
c 


We  find  in  the  table 

sin  48°  31'  =  .74915,      .•.  A  =  48°  31'  to  the  nearest  minute; 
B  =  90°  -  A  =  41°  29';    &  =  c  cos  A  =  246  X  .6624  =  162.95, 


b  =  Vc2  -  a8  =  162.95. 
TABLE  FOR  SOLUTION  OF  RIGHT  TRIANGLES. 


Give-u. 

Required. 

Formulas. 

1.  a,   b 

A,  B,  c 

tan  A  =  -, 
b 

B  = 

90°     -     Ay 

c  =  b  sec  ^1  =a  sec  .Z?. 

2.  a,    c 

A,  B,  b 

a 

sin  A  =  —  , 

B  = 

90°  -  A, 

5  =  c  cos  .4  =a  tan  B. 

3.  A,  a 

B,  b,  c 

B  =  90°  - 

Ay  b 

=  a  tan  B 

,  c—  <zsec  j5=&secu4. 

4.  A,  b 

B,  a,  c 

B  =  90°  — 

A,  a 

=  b  tan  A 

,  c=a  sec  _/?:=&  sec  A. 

5.        Ay      C 

B,  a,  b 

B  =  90°  - 

A,  a 

=  c  sin  A 

,  b=at&n.  B=ccosA. 

If,  in  the  second  case  above,  b  is  given  in  place  of  a,  then  a 
and  5  as  well  as  A  and  B  change  places  in  the  formulas.  However, 
A  =  90°  —  B  is  the  same  as  B  —  90°  —  A. 

17.  We  will  now  deduce  formulas  for  the  solution  of  oblique 
triangles. 

Draw  BD  in  Fig.  10  perpendicular  to  AC.  Then,  from  the  tri- 
angle ABD, 

BD  =  c  sin  A,     .     (29) 
and,  from  the  triangle  BCD, 
BD  =  a  sin  C.     .     (30) 

Also  CD  •=.  AC  —  AD  —  b  —  c  cos  A.     .     .    .     (31) 


PLAKE   TRIGONOMETRY. 

Equating  (29)  and  (30)  gives 

a      sin  A 

c  sin  A  =  a  sin  c,     or     —  =  -: — ~. 
c       sin  6 


Similarly,  or  by  analogy, 

a  _  sin  A 
b~  sin  B' 

From  the  figure, 


(32) 


b       sin  B 
and    c=^C'  J 


+  (CD)*. 
Substituting  for  BD  and  CD  from  (29)  and  (31),  we  have 

a'2  =  c1  sin2  A  +  62  —  2bc  cos  -4  -|-  c2  cos2  A\ 
or,  since  sin2  A  -(-  cos*  4  =  1, 

a».  —  £2  _|_  c->  _  2£c  cos  ^}     or    COB  A  ~ 
Similarly,  or  by  analogy, 

cos  B  =          ^C ,     and 


From  (29)  and  (31)  we  also  have* 

BD          c  sin  A 

tan  G  =  7=^  =  T- 


sin 


6  -  c  cos  A       b 

cos  A 


tan  5  = 


tan  J.  = 


sin  (7    . 

a 

- —  cos  U 

0 

sin  B 


(34) 


c 
a 

18.  Let  ABC,  Fig.  11,  represent  a  plane  triangle,  the  parts  be- 
ing represented  as  usual. 

Take  GE  =  CA,  and  draw  AD  and  EH  perpendicular  to  AE. 
We  have 

CAB  +  OEA  =  180°  -  (7  =  4  +  7?. 

and       £4#  =  (7^4  -  GBA  =  l(A  +B)  -  B  =  ±(A  -  B). 


FIELD-MANUAL   FOR   ENGINEERS. 


Also      GAD  =  90°  -  CAE,    and     CD  A  -  90°  -  (AEC '=  CAE). 
Hence  CD  =  AC  =  b. 

Now 

a  +  b      BD      AD      AEi&n  \(A  +B)      tan 


JEET 


.    .    (35) 


FIG  11. 


From  triangle  ABE, 


or 


BE  _  sin  BAE 
AB~  jsin  AEB' 

a  —  b  _  sin  \(A  —  B) 

c      ~  sin  \(A  +  B) 

In  triangle  ABD, 

BD  _  a  -f-  b       sin  BAD       cos  |(^4  —  B) 
AB  ~      c      ~  sin  ^IJW?  ~~  cos  ^(^4  +  B)' 


(36) 


•     (37) 


Eq.  (37)  divided  by  (36)  also  gives 


tan 


+  B) 


tan 


—  B)' 


(38) 


which  furnishes  another  demonstration  for  (35). 

A  slight  variation  of  the  above  solution  of  the  tangent  problem 
was  given  by  the  author  in  Vol.  I,  No.  1,  of  The  American  Math- 
ematical Monthly.  It  has  since  found  its  way  into  text-books  on 
trigonometry.  Still  another  solution  by  the  author  may  be  seen 
jn  Vol,  III,  No,  11,  of  the  same  journal, 


FLAKE   TRIGONOMETRY.  35 


SOLUTION  OF  PLANE  OBLIQUE  TRIANGLES. 

19.  There  are  three  cases. 

CASE  I.  —  In  this  case  two  of  the  given  parts  are  a  side,  and  the 
angle  opposite  ;  the  other  part  being  either  a  side  or  an  angle. 
Example  1.  —  Let  A,  a,  and  B,  Fig.  10,  be  given. 

C  =  180  -  (A  +  B). 
From  (32), 

sin  B  sin  C 

b  =  a  —.  —  —  ;    similarly    c  =  a  —  :  —  -T-. 
sm  A  sm  A 

Example  2.  —  A,  a,  and  b  given. 

(32)  gives       sin  B  =  sin  A  —  ;     then  the  table  gives  B. 
Now          C  =  180°  -  (A  +  B)  ;    and    c  =  a       --. 


CASE  II.  —  Given  two  sides  and  the  included  angle. 
Let  b,  c,  and  A  be  given.     We  have,  from  (34), 


sin  A 
tan  C  = 


Then 


cos  A 

c 


=  ISO8  —  (4  +  <7) ;     and    a  =  b  ~. 


Or,  from  (33), 

a  =  (&'  -f  e*  -  2bc  cos 


Then 


sin  B  =  sin  A  —  ,    and    C  —  180°  —  ( A  -f-  B). 


CASE  III.  —  Given  the  three  sides  a,  b,  and  c. 
Eq.  (33)  gives 


COS  J.  = 


36  FIELD-MANUAL  FOK   ENGINEERS. 

Then 

sin  B  =  sin  A-,     and     C  =  180°  -  (A  +  B). 


The  above  formulas  are  all-sufficient  for  all  practical  purposes. 
This  chapter  constitutes  a  complete  treatise  on  trigonometry, 

though  the  deductions  from  it  are  endless,  as  the  examples  in 

arithmetic  are  endless, 

TABLE  FOB  SOLUTION  OF  OBLIQUE  THIANGLES. 

(See  Fig.  10.) 


Given.         Required. 

Formulas. 

6.  A,B,a  C,  b,c 
7.  A,  «,  b  B,C,c 
8.  A,  b,  c  B,G,  a 

9.  a,  b,  c  A,B,C 

asinB 

a  sin  (A  -f  B) 

sin  A 
b  sin  A 

sin  A 
a  sin  (A  +  B) 

a 
sin  A 

sin  A 
b  sin  A 

tan  B   —  • 

-  —  cos  A 
b 

a*-(b-cy 

~    sin  B  ' 

(a+b-c}(a+c-b) 

2bc 
b  sin  A 

2bc 

a 

Use  the  first  form  for  vers  A  with  a  table  of  squares,  the  second 
without ;  they  are  the  best  formulas  known  for  this  case. 

The  following  are  the  best  formulas  known  for  the  area. 
10.  Area  lab  sin  C  —  %ac  sin  B  =  \bc  sin  A. 

It  is  never  necessary  to  compute  but  one  unknown  part,  and  in 
the  3d  case  none  at  all,  to  have  the  required  data  for  one  of  these 
equations  ;  and  the  computation  is  shorter  than  by  any  other 
formula. 

Observe  that  a  sin  C  is  equal  to  the  perpendicular  from  Bupon  b, 
b  sin  G  is  equal  to  the  perpendicular  from  ^1  upon  <(,  etc. 


PLANE    TLUGONOMKTRY.  37 


TRIGONOMETRIC  FORMULAS. 

tl.  sin  A       =  4/1  —  cos'J  A  — —  2  sin  ^A  cos  ±A. 

cosec  A 

12.  cos  A       —  \'\  —  sin*  A  =  —     -   =  cos2  IA  -  sin'2  \A 

sec  4. 

=  2  cos2  ^  -  1  =  1  -  2  sin2  \A. 

13.  tan  A      — ;  = r  =  cosec  2  A  —  cot  2 A 

cos  A        cot  .4 

/  1  -  cos  2.4  sin  2/1 


sin  2  A  1  +  cos  2^  ' 

14.  cot  ^4.       —  cosec  2  A  +  cot  24  —  the  reciprocal  of  any  expres- 

sion for  tan  A. 

15.  sec  A      —  --  —  the  reciprocal  of  any  expression  for  cos  A. 

cos  A 

16.  cosec  A   —  —  -  -  -  —  the  reciprocal  of  any  expression  for  sin  A. 

sin  A 

17.  vers  A    =  1  —  cos  A  —  2  sin2  \A. 

18.  exsec  A  —  sec  J.  —  1  =  —   -p. 

cos  4 

19.  sin  (.1  ±  B)  =  sin  .1  cos  5  ±  cos  A  sin  5. 

20.  cos  (A  ±  B)  =  cos  A  cos  .5  T  sin  A  sin  Z>. 

21.  sin  A.  +  sin  Z?  =  2  sin  |(4  +  5)  cos  \(A  —  B). 

22.  sin  A  —  sin  I?  —  2  cos  |(/1  -f  B)  sin  |(.l  —  7?). 

23.  cos  A  +  cos  B  =  2  cos  |(A  -f  .#)  cos  |(4  -  1?). 

24.  cos  B  —  cos  A  =  2  sin  \(A  +  5)  sin  £(J.  -  B). 

25.  sin2  4  -  sin2  B  =  cos2  B  -  cos2  /I  =  sin  (A  +  5)  sin  (.4  -  1?). 
20.  cos2  A  —  sin2  5=  cos  (4  -f-  B)  cos  (4  —  B). 

27.  tan  A  ±  tan  B=sin 

28,  cot  4  ±  cot  7?  = 


COS  4.  COS  5 

si"  <A  ±  B> 


38  FIELD-MANUAL    FOR   ENGINEERS. 

The  above  formulas  contain  the  practical  general  relations  exist- 
ing among  the  functions  of  an  angle.  By  writing  \A,  2A,  etc., 
in  place  of  A,  by  repetitious,  etc.,  the  formulas  may  be  greatly 
multiplied  without  producing  any  new  relations.  This  practice 
is  too  common  in  Trigonometries,  Field-books,  etc. 


CHAPTER   TV. 


SIMPLE  CURVES  CONNECTING  RIGHT  LINES. 

LET  ABODE  represent  a  circular  arc  joining  the  straigLt  lines 
A  V  and   E  V,   which  are 
tangent    to   the 


C' 


curve    at 
A  and  E. 

AV  and   JSV  are    tan-    V 
t^ents  to  the  curve,  A  and 
Z£are  tangent  points,  and 
the  angle  KVE  is  the  an- 
gle   of    intersection,    and 
shows  the  change  of  direc-    c 
tion  in   passing  from  one 
tangent  to  the  other. 

Vis  the  point  of  inter- 
section,  or  vertex. 

PROPERTIES  RELATING 
TO  THE  CIRCLE. 

The  following  proposi- 
tions rest  upon  elementary 
geometrical  principles, 
may  be  regarded,  for  the  most  part,  as  axiomatic. 

(a)  A  tangent  to  a  circle  is  perpendicular  to  the  radius,  at  the 
point  of  contact. 

(&)  Tangents  drawn  to  the  circle  from  the  same  point  are 
equal,  and  the  angle  between  these  tangents,  and  the  chord  join- 
ing the  tangent  points,  are  equal.  Thus, 

AV  -  EV,     and     VAE  -  VEA. 

(c)  The  central  angle  AOE,  subtended  by  a  chord,  is  equal  to 
the  exterior  angle  KVE  between  two  tangents  to  the  curve  at  the 
extremities  of  the  chord. 

(d)  The   angle   between   a  tangent  and  chord  is  equal  to  the 
angle  subtended  at  the  circumference  by  the  same  or  by  an  equal 
chord,     Thus,  VAB  -  BCA  =  BAG,  etc. 

39 


40 


FIELD-MANUAL   FOR    ENGINEERS. 


(e)  An  angle  between  a  tangent  and  chord,  or  an  angle  subtended 
at  the  circumference  by  that  chord,  is  equal  to  one  half  the  central 
angle  subtended  by  the  same  chord.  Thus, 


(/)  Equal  chords  subtend  equal  angles  at  the  center  of  a  circle, 
and  also  at  the  circumference.     Thus,  if  AB  --  BC,  etc., 

ACS  =  BAG,  etc.,     and    AOB  =  BOG,  etc. 


7 


FIG.  13. 


radius    perpendicular   to  a  chord  bisects  the  chord,  and 
also  the  angle  and  the  arc  subtended 
by  the  chord.     Thus,   if   OG  is  per- 
Px_^_ xN.T  pendicular  to'AE,  we  have 


AM  =  ME,     AOV  =  EO  V, 

and  AC=GE. 

(h}  Parallel  chords,  or  a  tangent 
and  parallel  chord,  intercept  equal 
arcs.  Thus,  in  Fig.  13,  if  PT,  P'T', 
and  P"I "  are  parallel,  then  PP" 
and  T'T"  are  equal,  also  PP'  and 
TT'  are  equal. 
(i)  The  exterior  or  deflection  angle  KBG,  Fig.  14,  between  any 

two  chords  AB  and  BG  is  equal  to  half  the  central  angle  AOG, 

subtended  by  the  chords.     This 

is  easily  shown  as  follows  : 

1.  Join   AC.      BAG  =  %BOC,  /     \i 
and   BGA   =  %AOB,    as    stated 

above.     But  /  / \C 

KBC=  BAG+BCA- 
.-.    KBC  =  ^AOB  +  iBOC 
=  \AOC. 

2.  Draw   the    tangent    EBL. 
Then 

KBC  =  KBL  +  LEG 
=  NBA  +  LBG 

Fio.  H. 


SIMPLE    CURVES    CONNECTING    RIGHT    LINES.          4l 

J3.  Since  the  chords  ^47?  and  BC  are  parallel  to  tangents  drawn 
at  the  'middle  of  the  arcs  AB  and  B C,  it  is  evident  that,  in  passing 
from  one  chord  to  the  other,  we  turn  through  an  angle  measured 
by  one  half  the  arc  AB  -\-  one  half  the  arc  BC,  that  is^  through  an 
angle  equal  to  one  half  AOC. 

SOME  ELEMENTARY  RELATIONS. 

In  Fig.  12  drop  the  perpendiculars  Bb,  Cc,  etc.,  upon  the  tangent 
AY. 

Ab,  be,  etc.,  are  called  tangent  distances,  and  Bb,  Cc,  etc.,  tan- 
gent offsets. 

Prolong  AB,  making  Bd  = '  AB.     dC  is  called  the  chord  offset. 

Then,  from  the  preceding  article,  we  have 

CBd  =  ^AOC  =  AOB. 

Drop  the  perpendicular  Bp  upon  Cd.  This  bisects  the  angle 
CBd  and  the  base  Cd.  Hence 

CBp  =  dBp  =  iCBd  =  \AOB  =  BAb. 

Since,  therefore,  the  triangles  CBp,  dBp,  and  BAb  have  an 
acute  angle  in  each  equal,  and  the  hypothenuses  also  equal,  they 
are  equal  in  all  respects  ;  and  therefore 

Cp  =  pd  =  Bb  ;     also     Cd  =  2Bb. 

Since  Ab  is  tangent  to  the  curve  at  A,  Bp  is  likewise  tangent  to 
the  curve  at  B. 

Represent  the  radius  AO  by  R,  the  tangent,  or  vertex  distance, 
A  Fby  T,  and  the  angle  of  intersection  K  VE,  as  well  as  the  central 
angle  AOE,  by  V.  Represent  the  chords  AB,  BC,  etc.,  by  c,  a 
long  chord,  as  AC,  by  C,  the  tangent  distances  Ab,  be,  etc.,  by 
d,  di ,  etc.,  and  the  tangent  offsets  bB,  cC,  etc.,  by  t,  ti ,  etc.  Then 
the  chord  offset  Cd  =  2t.  Represent  the  vertex  distance  GVby 
E,  and  the  middle  ordinate  of  a  long  chord  by  M.  Let  L  represent 
the  length  of  the  curve,  P. C.  (Point  of  Curve)  the  beginning  of 
the  curve,  and  P.T.  (Point  of  Tangent)  the  end  of  the  curve. 

The  degree  of  a  curve  has  been  defined  as  the  number  of  degrees 


FIELD-MANtJAL   FOR 


subtended  at  the  center  by  a  chord  100  feet  in  length.  This  defi- 
nition of  curvature  is,  however,  awkward,  arbitrary,  and  false. 
It  is  founded  on  error;  it  involves  unnecessary  labor  and  ends  in 
anomalous  and  erroneous  results. 

The  degree  of  a  curve  may  be  defined  as  the  change  in  its  di- 
rection between  one  point,  and  another  100  feet  from  the  first, 
measured  on  the  curve.  This  is  the  change  in  direction  which 
one  would  make  in  moving  100  feet  on  the  curve  from  one  point 
to  another.  Or,  as  the  angle  subtended  at  the  center  of  the  curve 
liy  an  arc  100  feet  in  length. 

Let  D  ==  the  degree  of  the  curve. 

The  circumference  of  a  1°  curve  is  therefore  360  X  100  =  36000 

OfjAAA 

feet,  and  its  radius  is  ~— —  —  5729.578  feet,  almost  exactly. 

Since  a  2°  curve  changes  its  direction  2°  in  100  feet,  its  circum- 
ference is  only  half  that  of  a  1°  curve ;  and  since  the  radius  varies 
directly  with  the  circumference,  the  radius,  too,  is  only  half  that 
of  a  1°  curve.  For  the  same  reason  the  radius  of  a  3°  curve  is 
precisely  £  the  radius  of  a  1°  curve;  and,  generally,  the  radius  of 
a  D°  curve  is  exactly  equal  to  the  radius  of  a  1°  curve  divided 

57*^9  57S 
by  1);  it  is  therefore  equal  to  1~ — . 

Referring  to  Fig.  15,  we  see  that  for  the  same  central  anglo 
AOE,  the  arc,  the  tangent,  the 
chord,  the  middle  ordinate,  the  er.- 
ternal  secant,  etc.,  vary  directly  with 
the  radius  or  inversely  with  the  de- 
gree of  curvature.  For  example  •. 
If  aO  =  %AO,  then  ape  -  \APE, 
ai  =  \AY,  etc.  If  APE  is  a  1°  or 
2°  or  3°  curve  of  length  L,  then  ape 
is  a  2°  or  4°  or  6°  curve  of  length 

Hence  to  compute    any  function 
of  any  curve,  the  tangent  or  exter- 
0  nal,   for  example,   from    the  corre- 

FIG.  15.  spending  function  of  another  curve, 

it  is  only  necessary  to  multiply  or  divide,  as  the  case  may  be,  by 
the  ratio  of  their  radii  or  degrees  of  curvature. 

Example. — Find  the  tangent  of  a  7°  13'  —  433'  curve,  the  cen- 
tral angle  being  37°  50', 


SIMPLE    CUHVES   CONNECTING  EIGHT   LINES.          43 

The  tangent  of  a  1'  curve,  by  Table  VII,  is  117812.     Then 

T=  117812  H-  433  =  272.1. 
Using  a  table  giving  functions  of  a  1°  curve,  we  have: 

Tangent  of  a  1°  curve  =  1963.6. 
Then        Tangent  of  a  1'  curve  =  1963.6  X  60  =  117816. 

Finally,  :i!7816  •+•  433  =  272.1. 

Or,  13'  T*|  60  =  0°.216, 

arid  therefore  7°  13'  =  7°. 216. 

Then  1963.6  -4-  7.216  =  272.1. 

With  a  table  giving  functions  of  a  1°  curve  there  is  no  escape 
from  dividing  by  60,  which  division  is  obviated  by  giving  the  func- 
tions of  a  1'  curve  as  in  Table  VII. 

It  is  often  desirable  to  know  the  difference  in  the  lengths  of  a 
chord  and  its  subtended  arc,  and  for  this  purpose  we  deduce  the 
following  formula  : 

d  =  .001269239249^*  -  .0000000048329—.  (8) 

nA  nb 

IP  D4 

=  .001269 .0000000048^,    very  nearly,  (9) 

ns  n5 

Z>2 
=  .001269—,    approximately (10) 

In  these  equations  D  =  the  degree  of  curvature,  d  =  the  differ- 
ence between  any  chord  and  the  subtended  arc,  and  n  =  arc  of  100 
feet  divided  by  this  arc. 

For  an  arc  of  100  feet  n  =  1  and  d  =  .001269Z*2,  nearly;  .  (11) 
For  an  arc  of  50  feet  n  =  2  and  d  =  .000158D2,  nearly;  .  (12) 
For  an  arc  of  25  feet  n  =  4  and  d  =  .00002Z)'2,  nearly.  .  (13) 

For  n  sub-chords  per  station  the  sum  of  the  difference  per 
station  is 

nd  -=  .001269  — (14) 


44 


FIELD-MANUAL   tOlt   ENGINEERS. 


Representing  the  central  angle  of  the  curve  l>y  V,  and  tlie  num- 
ber of  stations  in  the  curve  by  A",  we  have  N  ;  and   hence  tin- 

length  of  the  curve  exceeds  the  sum  of  the  lengths  of  the  sub- 
chords  by 

7)2        V  VD 

E=  Nnd=  .001269=^  X  -~  =  .001269—,..      .     (14') 


Hence,  in  laying  out  curves,  —  should  be  nearly  constant. 

Since  in  a  4°  curve  the  chord  of  an  arc  of  100  feet  is  99.98  feet, 
curves  from  0°  to  4°  can  be  properly  laid  out  with  chords  of  100 
feet,  and  with  the  same  degree  of  accuracy  we  may  lay  out  curves 
from  4°  to  16°  with  chords  of  50  feet,  and  those  from  16°  to  04° 
with  chords  of  25  feet. 

Very  sharp  curves  can  be  easily  laid  out  by  swinging  a  chain 
around,  while  one  end  is  held  at  the  center  of  the  curve. 

Let  s  represent  any  arc,  c  its  chord,  and  et  the  chord  of  one  half 
of  the  arc  s. 

Then,  from  above, 


This  is  said  to  be  Huygens'  approximation  to  the  length  of  an 
arc. 

The  following  table  shows  the  differences  between  arcs  of  25 
feet  and  of  50  feet,  and  the  chords  of  those  arcs.  (See  Fig.  16.) 


De* 

D 

Arc 

25 

Arc 

50 

Deg. 
D 

Arc 
s>5 

Arc 

50 

Deg. 
D 

Arc 

25 

Arc 

50 

1 

.000 

.000 

11 

.002 

.019 

21 

.000 

.070 

.000 

.001 

13 

.003 

.023 

22 

.010 

.077 

3 

.000 

.001 

13 

.003 

.027 

23 

.010 

.084 

4 

.000 

.003 

14 

.004 

.031 

24 

.011 

.091 

5 

.000 

.004 

15 

.004 

.036 

25 

.012 

.699 

6 

.001 

.000 

16 

.005 

.041  . 

26 

.013 

.107 

7 

.001 

.008 

17 

.006 

.046 

27 

.014 

.116 

8 

.001 

.010 

18 

.COS 

.052 

28 

.01C 

.124 

9 

.002 

.013 

19 

.007 

.058 

29 

.017 

.133 

10 

.002 

.010 

20 

.008 

\m 

30 

.018 

.143 

bIMI'LE    CURVES    CONNECTING    KKIHT    LINES. 


45 


II.   Suppose  the  chord  AB  =  100  feet,  and  AOB  =  D,  (Fig.  16). 
Then 

AOM=^,  MAtt  =  —  ,  etc. 

2  4 

Hence 
AM  =  AEsec  MAE 

=  50  sec  ~i. 

4 
Therefore 


AM  -  50  =  50 (sec  -£  -  1  j 


=  50  exsec  — . 

Similarly,     AK  =  AFsvc  —  =  25  sec  — -  sec  — ,  etc.,  etc. 
o  4  o 

The  following  table  gives  the  excess  of  AK  over  25  feet,  and  of 
AM  over  50  feet,  when  chord  AB  is  100  feet. 


Deg. 

AK 

AM 

Deg. 

AK 

AM 

Deg. 

AK 

AM 

D! 

-25 

-50 

A 

-25 

-50 

DI 

-25 

-50 

1 

.000 

.000 

11 

.036 

.058 

21 

.131 

.211 

2 

.001 

.00-2 

1-2 

.043 

.069 

22 

.144 

.231 

3 

.003 

.004 

13 

050 

.('81 

23 

.157 

258 

4 

.005 

.008 

14 

.058 

.093 

24 

.171 

.275 

5 

.007 

.012 

15 

.067 

.101 

25 

.186 

.29?. 

6 

.011 

.017 

16 

.076 

.122 

26 

.201 

.323 

7 

.015 

.0:23 

17 

.086 

.138 

27 

.217 

.349 

8 

.019 

.030 

18 

.096 

.155 

28 

.233 

.375 

9 

.024 

.039 

19 

.107 

.172 

29 

.250 

.408 

10 

.030 

.048 

20 

.119 

.181 

30 

.268 

.431 

Comparing  the  preceding  tables  we  learn  that,  when  the  chord 
AB  is  100  feet,  the  chord  AM  differs  nearly  four  times  as  much 
from  50  feet  as  the  chord  of  the  arc  of  50  feet  differs  from  50  feet. 
Furthermore,  that  the  chord  AK  differs  nearly  sixteen  times  as 
much  from  25  feet  as  the  chord  of  the  arc  of  25  feet  differs  from 
25  feet. 

Thus  let  us  first  suppose  A  MB  (Fig.  16)  to  be  a  10°  curve,  and 
the  chord  ^47?  =  100  feet.  Then  by  the  table  AK  =  25.030  feet, 
and  this  would  lead  to  an  error  of  .030  X  40  —  1.2  feet  in  laying 
out  a  curve  25  X  40  =  1000  feet  long,  taking  chord  AK  =  25  feet 
long. 


46  FIELD-MANUAL  FOR  ENGINEERS. 

Suppose,  secondly,  that  the  are  AMD  —  100  feet  and  therefore 
arc  A K  —  25  feet,  and  the  chord  AK  is  but  .002  of  a  foot  less  than 
25  feet. 

Hence  in  laying  out  the  above  curve,  taking  the  chord  AK '  —  25 
feet,  the  error  committed  would  be  only  .002  X  40  =  .08  of  a 
foot. 

Thus  we  see  that  when  the  chord  of  a  station  is  taken  =  100 
feet,  the  sub-chords  AM,  AK,  etc.,  differ  so  much  from  50  feet, 
25  feet,  etc.,  as  to  largely  vitiate  the  results,  whereas  such  is  by 
no  means  the  case  when  the  arc  AMB  is  taken  =  100  feet. 

Indeed,  when  the  chord  AB  is  100  feet,  the  shorter  the  sub- 
chords  used  in  laying  out  a  curve,  the  greater  the  discrepancy  in 
the  measurement,  on  this  basis.  Thus  for  a  10°  curve  the  difference 
between  four  equal  sub-chords  and  100  feet  is  .080  X  4  =  .120  of  a 
foot ;  whereas  the  difference  between  two  equal  sub-chords  and 
100  feet  is  .048  X  2  =  .096  of  a  foot.  The  reverse  is  of  course  the 
case  when  the  arc  AMB  is  made  the  standard  of  measurement. 

These  facts  are  evident ;  for  when  the  chord  is  made  the  stand- 
ard of  measurement,  the  sum  of  the  lengths  of  the  sub-chords  ex- 
ceeds more  and  more  the  length  of  the  chord,  the  shorter  they  are  ; 
whereas  when  the  arc  is  the  standard  of  measurement,  the  sum  of 
the  lengths  of  the  sub-chords  falls  short  of  the  length  of  the  arc 
less  and  less  the  shorter  they  are. 

Most  recent  writers  have  endeavored  to  obviate  the  inconven- 
iences and  inconsistencies  above  pointed  out  by  inconsistent  assump- 
tions, such  as  basing  the  curves  of  different  degrees  upon  chords 
of  different  lengths.  This  scheme  gives  the  values  of  some  of  the 
radii  quite  correct ;  but  it  causes  sudden  breaks  in  the  value? 
where  the  changes  are  made. 

For  example,  how  can  there  be  two  different  values  (819.02  an<* 
818.64)  of  the  radius  of -a  7°  curve? 

And  why  should  the  radius  of  a  7°  10'  curve  be  given  quite  cor- 
rect, while  that  of  a  6°  50'  curve  is  quite  incorrect? 

Again,  we  are  told  by  a  recent  author  that,  in  practice,  it  is  cus- 
tomary to  take  the  radius  of  a  1°  curve  as  5730  feet,  and  to  assume 
the  radius  to  vary  inversely  as  the  degree.  Thus  for  a  4°  curve 

the  radius  would  be  — j-  =  1432.5  feet.     This  is   rational   and, 
4 

moreover,  is  precisely  what  is  here  advocated,  except  that  the  true 
value  of  the  radius  of  a  1°  curve  is  used,  vi/.,  5729.58  feet. 


SIMPLE   CURVES   CONNECTING    RIGHT   LINES.         47 

FORMULAS. 
From  the  triangle  AOV,  Fig.  12,  we  have 

AV=AOt&nAOV,     or     T=Rtau^V.     .     .     (15) 
Transposing,  we  Lave 


From  the  triangle  AOF,  we  find 


A0  =     .Ani;,    or    R  =  -rn  =  &  cosec 
sin  ^107^  sin  \J) 

This  value  of  R  in  (15)  gives 


Measure  equal    distances  FIT"  and   VL  along  the   tangents  in 
Fig.  12  and  draw  HNL. 
Measure  #JVand  VN. 

A  V        VN  VN 


that  is,  T=R-.   ...  .....     (19) 

If  ^4J/  and  FJf  are  measured,  then 


Eqs.  (19)  and  (20)  serve  to  fix  geometrically,  without  measuring 
angles,  the  tangent  point  of  a  curve  of  given  radius  that  will  unite 
two  straight  lines  on  the  ground. 

In  Fig.  12  draw  VG'  perpendicular  to  A  V  to  meet  A  C  prolonged. 
Now  the  angle  AVG  —  \AVE  =  |(180  -  F)  =  90  -  |F.  Hence 

VCC'  =  AGO  =  90°  -  BOO  =  90°  -  4LF. 


48  FIELD-MANUAL    FOR    EXGJNKEkS. 

Also,   VG'C  =  90°  -  CA  V  =  90°  -  i  V; 

.-.     VC'C-  VCC',     and     VC'  =  VG  =  #. 
Hence 

7^7  /"fr   TT 

^-  =  -jy  =  laniF,     or     E  =  T  \&\\  ±V.    .     .     ("21) 

Substituting   T  —  ft  tan  IF  for  T,  from  (15),  gives 

^=  tftan^Ftan  £F.    ......    (21') 

Since  the  triangles  BCd  and   7?0(7are  similar,  \ve  have 
Gd        BC  BC* 


or 


and  Bb  =  t  =     ~  .........  (3:.) 

We  also  have 

56  =  AB  sin  7?^4&,     or     i  =  c  sin  |D,    .     .     .  (24) 

and  2t  =  2c  sin  \D  ........  (25) 

Example.  —  Given  R  =  1909.9  feet  and  c  —  100  feet,  to  find  the 
tangent  and  the  chord  offsets  for  100  feet. 
By  (23), 

1  0000 

=  2.618,    and    2t  =  5.236, 


or  Table  I  shows  that  1909.9  is  the  radius   of   a  3°  curve,   and 

\D  =  1°  30',     and     sin  \D  —  .02618. 
Then,  by  (24), 

t  =  .02618  X  100  =  2.618,     and    2t  =  5.236. 

The  tangent  offsets  are  given  in  Table  I,  and  the  chord  offsets 
are  twice  the  tangent  offsets. 


SIMPLE    CURVES    CONNECTING    RIGHT    LINES.          49 

In  laying  out  curves,  the  chain  is  stretched  from  point  to  point 
>n  the  curve,  and  coincides,  therefore,  with  chords  of  the  curve. 

Since  the  process  is  the  same  whatever  the  length  of  chain 
used,  we  will  assume  it  to  be  100  feet  long1. 


FIG.  17. 

The  length  of  the  curve  is  expressed  in  chains,  in  terms  of  the 
central  angle  AOF  =  V,  and  the  degree  of  the  curve  AOB  =  D. 

V 
The  number  of  chains  is  evidently  equal  to  —  . 

Thus,  in  the  figure,  if  V—  23°  and  D  =  5°,  the  curve  is  -2/  =  4f 
chains,  or  460  feet  long.  As  the  angle  AOE  =  5°  X  4  —  20°, 
EOF—  23°  —  20°  =  3°  ;  and  the  arc  EF  is  f  X  100  =  60  feet  long. 

It  is  usual,  in  laying  out  curves,  to  assume  the  radius  11,  and  to 
find  the  degree  of  the  curve  D  from  it  ;  or  to  assume  D  (usually 
in  degrees  and  minutes),  and  to  find  R  from  it.  Neither  way 
is  best. 

To  assume  a  value  of  R  or  of  D  does  not  aid  in  the  least  in 
properly  locating  the  curve.  Generally  the  surface  of  the  ground 
does  indicate  approximately  the  position  of  the  curve,  and  the 
proper  course  to  pursue,  therefore,  is  the  following:  Divide  the 
tangent  of  a  one-minute  curve  by  the  length  in  round  numbers  of 
the  desired  tangent,  and  neglect  the  decimal  in  the  quotient. 
This  give.s  the  degree  of  the  required  curve  in  minutes.  We  may 


..nzrociTV     11 


50 


FIELD-MANUAL    FOR    ENGINEERS. 


change  the  quotient  to  an  even  number  of  minutes,  or  to  some 
multiple  of  10  if  we  wish,  if  there  is  sufficient  latitude  to  be  taken 
in  the  position  of  the  P.C. 

Then  divide  the  tangent  of  a  one-minute  curve  by  the  corrected 
quotient  for  the  tangent  required. 

Example.  —  V—  46°  30',  and  the  tangent  should  be  1100  feet  or 
over.  Find  the  degree  of  the  curve,  and  the  length  of  the  tan- 
gent. 

Dividing  the  tangent  of  a  1   curve  by  1100  gives 

147697  +  1100  =  134'  =  2°  14'. 

Now  2°  10'  =  130',  and  147697-^-  130  =  1136.13,  the  tangent  re- 
quired. 

LONG  CHOKDS  AND  ORDINATES  TO  LONG  CHORDS, 

Let  A,  B,  C,  etc.,  represent  stations  upon  a  curve. 

Draw  the  lines  as  repre- 
sented, EX  being  a  perpendic- 
ular from  the  middle  of  the 
curve  E  upon  the  tangent 
AT.  Let  Jlf  =  EM,  M'  =  FN, 
etc. 

Draw  AE,  and  we  have  the 
angle  EA  M  equal  to  the  angle 
EAX,  and  therefore  the  tri- 
angles EAM  and  EAX  are 
equal  in  all  respects.  From 
this  we  learn  that  the  tangent 
offset  EX  for  any  arc  AE  h 
equal  to  the  versin  EM  of 
that  arc,  or  to  the  middle  or- 
dinate  EM  ot  the  chord  AK 
of  twice  the  arc. 

Or,  draw  the  tangent  Et  and  the  perpendicular  Kt  upon   it. 
Then  the  tangent  offset  Kt  of  the  arc  EK  =  the  versin  EM  of 
the  same  arc  =  the  middle  ordinal  e  EM  of  the  arc  AEK. 
To  find  the  middle  ordinate,  M.     (See  Fig.  18.) 


1°.  M  =  EM  =  EX  =  AM  tangent  EAM  =  \C  tan 
3°.  M  =  EM  =  EX-  ET  cos  TEX  -  #eos    V, 


.      (26) 
.     (26'  ) 


SIMPLE   CURVES   CONNECTING    RIGHT  LINES.          5l 

3°.  M  =  EO  -  MO  =  U  -  R  cos  |  F  =  #(1  -cos  £  F) 

=  R  vers  |  F.  .     .     .    (26") 

4°.  J/=  EO-MO=EO-  i/AO*-AM*=R-  \/ltr-iC-.   .  (26'") 

To  find  any  ordiuate  FN  distant  d  from  the  center  of  the  chord. 
Prolong  FN,  to  meet  OS,  drawn  parallel  to  AK,  and  join  FO. 


Now,     FS  = 


-  OS"  =  4/-R8  -  d*.  JVS  =  MO  =  R  -  M  . 


=  M+ 


-  -  d*  -  R. 


Other  methods    will   be   given    in  connection  with  laying  out 
curves  by  ordinates  from  a  long  chord. 


Approximate   Values  of  Ordinates  to  Short  Chords, 

^FG, 


Divide  the  chord  into  any  number 
of  equal  parts,  eight  for  example,  at 
z,  k,  ly  etc. ,  and  erect  the  ordinates 
Em,  Fn,  etc.,  and  prolong  them  to  A 
meet  the  curve  in  13',  F',  etc.  Let 
Em  =  m,  Fn  =  m',  etc.  We  have, 
from  geometry, 

_  Am  X  mB  __  \c  X  $c 
or,  approximately, 


»»=   2r-=55-  •  •  <27> 

6.K  oil 


E'F 
FIG.  19. 


^G 


JW 


Similarly, 


c  X  |c      15     c* 


^L 
WSR 


=  m    =-          =       .  ~  =      m  =  m  - 


Hq  =  m'"  = 


(28) 

(28') 
(28") 


For  any  other  equal  divisions  of  the  chord  we  have  similar  re-, 
suits. 


FIELD-MANUAL    FOR    ENGINEERS. 

tablisli  the  points  E,  F,  etc. 

Set  E  equally  distant  from  A  and  13,  and  at  the  distance  m  from 
the  point  m  ;  then  F,  equally  distant  from  k  and  13.  and  \im 
from  n  ;  then  G,  equally  distant  from  m  and  B,  and  £w*  from  p; 
lastly,  H,  equally  distant  from^)  and  B,  and  T7gw  from  q. 

This  method  involves  much  less  labor  than  that  of  drawing  sub- 
chords  to  find  the  points  F,  G,  etc.  If  we  draw  a  tangent  at  E, 
the  offsets  to  F,  G,  H,  etc.,  will  be,  according  to  the  preceding 
formulas,  fam,  yV^,  ~i%m>  e^c-  This  shows  a  convenient  way  of 
finding  the  points  on  the  curve. 

To  compute  tangent  offsets  and  middle  ordinates  by  means  of  a 
series. 

In  a  way  similar  to  that  pursued  in  finding  the  difference  be- 
tween an  arc  and  its  chord  we  find 

t  =  .872664625997Z>  -  .  000022 15240389.D3 

-f.00000000022493360386Z>5 -,  .     (29) 

or 

t  =  .872664626,0  -.000022152404£3-f-.000000000224933604Z)5.  (30) 

We  may  find  the  tangent  offset  for  m  stations  by  multiplying 
the  successive  terms  of  (30)  by  m1,  m4,  etc. 

We  thus  find  tang  offset  for  arc  of  50  feet,  or  —  stations, 

2 

t1  =  .2181G6156499Z)  -  . 000001 38452524D3 

+  .00000000000351458756Z)5-.     .     (31) 

For  1  foot  m*  =  (.Ol)'2  =  .0001,  m4  =  .00000001,  etc.     Then, 

The  tang  offset  for  1  ft. 

=  t,  =  .0000872664626Z)  -  .00000000000022152404D3  +  .  (32) 

From  (30),  we  have  t  =  .SID,  approximately. 

For  n  stations  tn  =  .87/iJX> (33) 

Wellington  in  Railway  Location,  ch.  xxx,  recommends  the 
equation  t  =  |»8A  ,  ,  , (33') 


SIMPLE   CURVES   CONNECTING   RIGHT  LINES.          53 

\vliicli  is  practically  the  same  as.  (33).  (33)  is  a  trifle  more  accurate 
than  (33'),  but  either  is  sufficiently  accurate  for  all  cases  in 
which  ii-D  does  no*  much  exceed  thirty. 

The  same  formuia  gives  the  middle  ordinate,  n  representing  the 
number  of  stations  on  either  side  of  the  center,  or  2n  the  number 
of  stations  in  the  arc.  Any  other  ordinates  desired  are  then  given 
by  eq.-?.  (27).  (28),  etc. 

Example.—  Find  six  offsets  of  a  3°  curve  at  points  50  feet  apart. 
(See  Fig.  18  ) 


For 

n 
n 
n 

n 
n 
n 

= 

1, 
1, 

t. 
2, 

2i, 
3, 

^    —   7 
t  = 

t   = 
t  — 

•  x 

X 
X 

\s 

x 

i 

1 

9 

•r 
4 

L'  5 
"T" 

6 

X 

x 

X 
X 

x 

X 

3 
:j 
3 
8 
3 
8 

1= 

9 

5 

10 

16 

2:' 

.66; 
.62; 
91; 
.50; 
.41; 
.62. 

These  results  are  of  course  equal  to  the  middle  ordinates  for 
1,  2,  3  ...  6  stations  of  the  same  curve. 

LAYING  OUT  CURVES. 

Since  in  laying  out  curves  the  operation  or  method  is  the  same 
whatever  the  length  of  the  chain  or  chord,  we  will  here  assume  it 
to  be  100  feet  long. 

A.     By  Deflection  Angles. 

Let  A  in  Fig.  17  be  the  P.C.  Set  the  instrument  at  A,  and 
turn  off  from  the  tangent  AVthe  given  deflection  angle  VAB  = 
\D,  D  being  the  degree  of  curve.  This  will  give  tlie  direction, 
AB,  and  measuring  100  feet  in  this  direction,  the  point  B  will  be 
determined.  Turn  off  the  additional  angle  B  AC  =  \T),  the  tele- 
scope being  now  directed  toward  C,  and  set  C  in  the  line  AC  &nd 
100  feet  from  B.  Turn  off  the  additional  angle  CAD  =  ^D,  and 
set  D  in  the  line  AD  and  100  feet  from  C.  Proceed  in  the  same 
way  for  other  stations  to  the  end  of  the  curve,  or  so  far  as  the 
stations  can  be  seen  from  A. 

It  is  usually  impossible,  on  account  of  obstructions  likely  to  be 
met  with,  to  lay  out  the  whole  of  a  curve  from  the  first  station. 
When  such  is  the  case,  we  determine  as  many  stations  as  conven- 
ient, remove  the  instrument  to  the  last  station  so  determined,  and 


54  FIELD-MANUAL    FOR   ENGINEERS. 

proceed  from  that  as  from  the  first  station.  For  example  :  Suppose 
B,  C,  and  D  to  be  found  with  the  instrument  at  A.  Remove  the 
instrument  to  Z),  sight  to  A,  turn  off  the  angle  ADV  '  =  DA  V  — 
\D,  and  the  line  of  sight  will  be  in  the  direction  of  the  new  tan- 
gent DV  Sit  D.  Reverse  the  telescope,  and  the  line  of  sight  will 
point  forward  along  the  same  tangent  VDG.  Now  set  E,  F,  etc., 
from  the  tangent  DG,  as  B,  C,  and  D  were  set  from  the  tangent 


In  setting  the  first  stake  from  the  new  position  of  the  instru- 
ment, as  E  from  D,  no  notice  need  be  taken  of  the  tangent  at  D, 
it  being  necessary  simply  to  turn  off  from  the  line  ADUihe  angle 
HDE  =  HDG  -f  GDE  =  AD  V+  GDE  -  4(iZ>). 

In  the  new  position  of  the  instrument  we  observe  that,  in  all 
cases,  the  deflection  from  the  line  pointing  to  the  back  station  to 
the  line  pointing  to  the  forward  station  is  as  many  times  the  de- 
flection angle  \D  as  there  are  chains  in  the  curve  between  the 
back  and  the  forward  station.  Of  course  this  applies  to  simple 
curves  only. 

The  beginning  of  a  curve,  as  well  as  the  end,  usually  falls 
between  regular  stations,  giving  short  chords  at  the  ends.  The 
deflection  angle  for  a  short  arc  is  such  a  part  of  the  full  deflection 
angle  as  the  short  arc  is  of  the  full  arc. 

Let  a  represent  the  arc  AB  or  BC,  etc.,  and  a,  represent  the 
arc  EF.  Also  EOF  =Di. 

Hence     DFE—^D,     and     EDF=$D1,     and  therefore 


—  =  —  ,     or 
\D        a 

In  beginning  a  curve,  the  instrument  being  at  the  first  station, 
it  is  convenient  to  place  the  zeros  of  the  instrument  plates  to- 
gether, and  direct  the  line  of  collimation  along  the  tangent  to  the 
curve.  The  reading  on  the  limb  for  any  station  will  then  be 
equal  to  the  total  deflection  angle  for  that  station. 

If  the  vernier  is  not  disturbed  while  laying  out  the  curve,  it  is 
plain  that  when  the  instrument  is  moved  to  its  second  position  D, 
Fig.  17,  and  the  line  of  sight  directed  to  A,  the  reading  will  be 
equal  to  the  total  deflection  from  A  to  D;  and  that,  an  additional 
angle  equal  to  this  deflection  being  turned  off,  the  reading  will  be 
equal  to  the  central  angle  AOD,  and  the  line  of  sight  will  be  in 
the  direction  of  the  tangent  at  D.  The  same  is  true  for  all  posi- 


SIMPLE    CURVES   CONNECTING   EIGHT   LINES.          ^ 

tions  of  the  instrument.  Since  the  deflections  for  the  last  section 
of  the  curve,  that  is,  from  the  last  position  of  the  instrument  to 
the  end  of  the  curve,  is  turned  off  but  once,  the  reading  on  the 
instrument,  when  the  curve  is  finished,  will  be  equal  to  the  total 
central  angle  V,  less  the  deflection  for  the  last  section.  This  fur- 
nishes a  convenient  check  for  the  work, 


B.  By  Tangent  Offsets.     New  Method. 

Let  ABCDEFGH  represent  a  curve  having  a  short  chord  AB 
--  c'  subtending   an  angle  AOB 
=  -  Di  at  one  end  of  the  curve. 

Define  the  tangent  A  V  by  set- 
ting stakes  upon  it.  Draw  Be, 
TX  etc.,  parallel  to  A  V,  and  BX, 
iJcX',  etc.,  perpendicular  to  AV, 
or  suppose  such  lines  to  be 
•drawn.  Now 

BX=ABsm£A  X=c'sm  \Di  =  t^ 

and  is  given  by  Table  II. 

Since  the  chords  BC,  CD,  etc., 
are  parallel  to  the  tangents  at  the 
middle  of  the  arcs  BC,  CD,  etc.,  FIG.  20, 

we  have 

CBc  =  D,  +  \D,     DCd  -  D,  +  \D,     EDe  =  A  -f-  ID,  etc. 
Hence 

Cc  =  c  sin  (D1  4-  |£),     Dd  =  c  sin  (D,  +  |Z>), 
Ee  =  c  sin  (D,  -f  |Z>),     etc. 

We  observe  that  Cc,  Dd,  Ee,  etc.,  are  respectively  the  tan- 
gent offsets  for  curves  of  the  degrees  (Di  -(-  ^D),  (Z>i  -f-  %D), 
(2>,  _|_  |/))>  etc.,  and  may  be  taken  from  Table  III.  Then 

CX'  =  BX  +  Cc,  DX"  =  CX'  +  Dd,  EX"'  =  DX'  +  Ee,  etc. 

Set  B  at  a  distance  c'  from  A,  and  ti  from  A  V;  then  C  a  dis- 
tance c  from  B,  and  CX'  from  AV;  D  a  distance  c  from  C,  and 
Dx"  from  A  V,  etc. 


FIELD-MANUAL    FOR   ENGINEERS. 


be  observed  that  in  this  method  we  avoid  constructing 
t  at  B,  in  consequence  of  the  short  chord  AE\  we  avoid, 
secondly,  the  finding  of  AX,  XX',  etc.;  thirdly,  the  errection  of 
perpendiculars,  at  X,  X',  etc.;  and,  lastly,  we  avoid  the  use  of 
the  radii  which  are  large  and  fractional.  Since  c  is  generally  100, 
though  sometimes  an  aliquot  part  of  100,  no  computation  is  gener- 
ally required,  and  none  to  mention  in  any  case.  Thus  we  see  how 
simple  and  short  this  method  is,  compared  with  the  method  of 
tangent  offsets  in  general  use. 

When  the  curve  begins  at  a  station  there  is  no  short  chord. 
Then 

A  B  =  c,  D!  —  D  ;  BX  =  c  sin  $D,  Cc  =  c  sin  |Z>,  Ed  —  c  sin  f  Z>, 

etc. 

It  is  best  to  lay  out  the  curve  from  each  end,  so  as  to  avoid  off- 
sets inconveniently  long.  Long  offsets  may  be  avoided  by  drawing 
a  tangent  at  any  station  and  continuing  the  curve  from  that  station 
precisely  as  from  A,  when  there  is  no  short  chord. 

To  draw  a  tangent  at  any  station,  draw  a  line  through  that  sta- 
tion and  at  a  perpendicular  distance  from  an  adjacent  station  equal 
to  the  tangent  offset  for  a  station  =  c  sin  \D.  Or,  draw  it  parallel 
to  the  chord  joining  adjacent  stations. 

It  will  be  noticed  that  stations  on  the  curve  are  not  opposite 
stations  originally  set  on  the  tangent. 

The  length  of  the  curve  gives  the  number  of  the  station  at  H. 

This  is  perhaps  the  best  of  all  methods  without  a  transit ;  but 
a  combination  of  methods  is  sometimes  advisable,  as  will  be  shown 
further  on. 

We  also  have 

AX—  CCOS^D!  ,  XX'  —  Be  =  c  cos  (Dl  -f  ^D),  etc., 

or  XX ',  X'X",  etc.,  are  respectively  equal  to  the  tangent  distances 
for  one  station  of  curves  of  the  degrees  D\  -f-  \D,  D\  -f-  f  Z),  etc. 

These  quantities  are  not  needed,  however,  and  it  is  to  be  ob- 
served that  the  points  X,  X',  etc.,  are  not  established  or  used. 


SIMPLE    CURVES    CONNECTING    EIGHT    LINES. 


C.   To  Locate  a  Curve  by  Ordinates  from  a  Long  Chord. 

Let  Fig-.  21  represent  a  curve  having  an  odd  number  of  full 
chords,  and  a  short  chord 
AB  —  c'  subtending  an 
angle  AOB  =  Dl}  and  a 
short  chord  KL  —  c"  sub- 
tending an  angle  KOL 
=  D<I.  Draw  the  tan- 
gent AT,  and  drop  the 
perpendiculars  BX  and 
CX'  upon  the  tangent. 
Draw  Be  parallel  to  AT. 

Since  Be  is  perpendic- 
ular to  AO,  and  BG  is 
perpendicular  to  a  radius 
bisecting  the  angle  BOG,  the  angle  GBc  =  J)l-\-  \D. 

Now  BX  is  found  in  Table  II  ;  and  Ce  may  be  found  from  the 
same  table  or  Table  III,  supposing  the  curve  to  be  of  the  degree 
A  +  \D.  Or, 

BX  —  c'  sin  IDj  ,     and     Cc  =  c  sin  (Dl  -f-  %D}. 
Suppose  c'  =  80,      D  =  4°. 


i=3°  12', 


and 

Table  II  gives 
and 


Cc  =  9.06. 
Y'  =  11.29. 


Place  B  80  feet  from  A  and  2.23  feet  from  the  tangent  A  T ;  then 
C  a  chain-length  from  B  and  11.29  feet  from  the  tangent  AT. 

Locate  K  from  the  tangent  at  L  in  the  same  manner  as  B  is 
located.  This  gives  the  line  CK. 

Suppose  the  stations  already  located  ;  the  long  chords  EG,  DH, 
etc.,  drawn,  and  also  the  perpendiculars  Dd,  Eel,  etc. 

Now  Cd=  Cm  -  Di  =  l(CK—  DH)  =  one  half  of  the  difference 
between  the  chords  of  six  and  of  four  stations.  See  Table  IV. 

dl  —  Di  —  Ef=  \(DH—  EG]  =  one  half  of  the  difference  be- 
tween the  chords  of  four  and  of  two  stations. 

lm  =•  one  half  the  chord  of  two  stations. 


58 


FIELD-MANUAL   FOR   ENGINEERS. 


Again,  Dd  —  Fm  —  Fi  =  the  difference  between  the  middle 
ordinates  of  chords  of  six  and  of  four  stations.  (See  Table  V.) 

El  —  Fm  —  Ff  '  =  the  difference  between  the  middle  ordinates  of 
chords  of  six  and  of  two  stations. 

Finally,  Fm  is  the  middle  ordinate  of  a  chord  of  six  stations. 

We  note  that  points  and  lines  on  the  right  of  Fm  are  of  course 
symmetrical  with,  corresponding  points  and  lines  on  the  left. 


Hence 


On  =  CK-  nK  =  CK  -  Cl 

Ch  =  CK-  hK  =  CK  -  Cd,  etc. 


Now  lay  off  Cd,  Cl,  Cm,  etc. 

Set  D  a  station  from  C  and  a  distance  Dd  from  d,  or  CK;  ther 
Ea.  station  from  D  and  a  distance  El  from  I,  or  CK;  then  F  i 
station  from  E  and  a  distance  Fm  from  m,  or  CK,  etc.,  etc. 

We  also  have 

Cd  =  c  cos  f  D,     dl  —  c  cos  f  D,     and     fan,  =  c  cos  \D  ; 
Dd  -  c  sin  %D,   Ee  =  c  sin  f  D,      and     .FJf  =  c  sin  £Z>. 

These  quantities  may  be  taken  directly  from  a  table  of  sines 
and  cosines,  as  already  pointed  out.  It  is  not  necessary,  as  shown 
under  the  preceding  method,  to  compute  and  lay  off  Cd,  Cl,  etc. 

D.     To  Locate  a  Curve  by  Chord  Offsets. 

Let  ABCDEF  be  the  curve,  having  short  chords  AB  =  c' ,  sub- 
tending an  angle  AOB  =  D\ 
and  EF  —  c",  subtending  an 
angle  EOF  =  D.,. 
Fa  Locate  B  as  in  the  las' 
T  method.  Prolong  AB,  mak 
ing  BCi  =  c.  The  angle  CBC, 
=  CAB  +  ACB  =  \D-\-\Di. 
Hence  CiCis  the  chord  offse 
2t'  for  a  curve  of  \D  -)-  \D 
degrees. 


=  2c  sin  i(Z>,  +  D}. 

Place  C,  therefore,  at  a  distance  c  from  B,  and  2t'  from  Ci,  2t'  beinjv 
twice  the  tangent  offset  which  is  given  by  Table  III  opposite  the. 


SIMPLE   CURVES   CONNECTING    RtGHf   LINES.          59 

degree  $(D  +  DI).  Prolong  BC  to  D,  ,  making  <7A  =  c.  Now 
D1D  —  2t  is  given  by  Table  III,  being  twice  the  tangent  offset. 
Place  D,  therefore,  at  a  distance  c,  from  (7,  and  2t  from  DI.  Place 
all  regular  stations  similarly.  C  may  be  placed,  also,  as  in  the 
preceding  method.  J^is  easily  placed  from  the  tangent  at  E,  as  B 
was  placed  from  the  tangent  at  A. 

It  may  be  observed  that  by  the  above  method  we  are  able,  by 
means  of  a  table  giving  offsets  for  full  stations  only,  to  locate  any 
station,  as  C,  by  chord  (or  tangent)  offsets,  though  the  chord,  as 
AB,  preceding  the  adjacent  chord  may  be  of  any  length. 

E.     To  Locate  a  Curve  by  Middle  Ordinates. 

Let  ABCDEFG  be  the  curve,  having  a  short  chord  AB  =  <•', 
subtending  an  angle  AOB  —  D\ , 
and  a  short  chord  FG  =  c",  sub- 
tending an  angle  FOG  =  D«. 

Locate  the  stations  B  and  C  from 
the  tangent  A x,  or  by  some  other 
method  as  already  explained. 
Then  set  off  on  CO  the  distance 
Cc  =  t  —  c  sin  \D  —  the  middle 
ordinate  of  a  chord  of  two  stations ; 
and  set  D  in  the  prolongation  of 
Be,  and  at  a  distance  c  from  C, 
Set  off  Dd  the  same  as  Cc,  and  set  E  in  the  prolongation  of  Cd, 
and  at  a  distance  c  from  D.  Locate  all  subsequent  stations  in  the 
same  way.  To  test  the  accuracy  of  the  work,  measure  the  per- 
pendicular Fy  from  the  last  regular  station  upon  the  tangent  at  G. 
This  distance  ought  to  be 


F.  To  Lay  out  a  Curve  by  Radial  Lines  from  the  Center. 

Consider  the  case  of  a  half-mile  race-track,  having  two  parallel 
sides,  each  600  feet  long,  connected  at  the  ends  by  semicircles,  as 

204Q 1900 

shown    in  Fig.  24.     Now    -    — — — '•  -  =  720,  the  length  of  each 

180° 
semicircle.     Hence  the  degree  of  each  curve  —  «-™  =  25°,  and 

the  radius  is  229.18  feet. 


66  FIELD -MAX  UAL    FOR   ENGINEERS. 

Set  the  instrument  at  0,  and  ran  radial  lines  01,  02,  etc.,  making 
6 


FIG.  24. 

angles  of  80°  with  each  other,  for  example,  and  set  stakes  at  1,  2, 
etc.,  229.18  feet  from  0.  These  stakes  are  100  f  <?  =  120  feet  apart, 
measured  on  the  arc. 

Determine  other  points  on  the  curve  by  middle  ordinates. 

The  inside  of  the  track  is  three  feet  inside  of  the  line  measured 
above. 

ERRORS  IN  FIELD-WORK. 

In  laying  out  curves,  as  well  as  in  all  field  operations,  errors 
will  sometimes  occur,  and  it  is  important  to  know  the  immediate 
and  the  ultimate  effect  of  such  errors;  to  know  when  such  errors 
are  increasing,  from  one  stage  of  the  work  to  another,  and  when 
they  are  decreasing;  when  they  are  too  small  to  be  of  importance, 
and  when  they  are  so  large  as  to  vitiate  the  result  if  not  cor- 
rected. 

It  is  important,  also,  to  know  the  law  of  increase  or  of  decrease 
of  such  errors;  for  in  that  case  the  error  at  any  point  may  be 
found  from  that  at  any  other  point,  the  end  of  the  curve,  for  ex- 
ample; thus  making  it  possible  to  properly,  that  is,  really,  correct 
the  curve  without  rerunning  it. 

I.   Curves  Laid  out  by  Deflection  Angles. 

A.  To  find  an  approximate  value  of  the  error  at  the  end  of  a 
curve  due  to  an  error  in  the  length  of  a  chord. — Suppose  A,  B,  C, 
etc.,  to  represent  stations  on  the  true  curve,  and  A',  B' ,  C",  etc., 
stations  on  the  false  curve. 

1°.  Suppose  the  first  stake  from  A  to  be  set  at  B' ,  a  distance 
BB'  =  e  beyond  B,  its  true  place. 


SIMPLE    CUliVK-i    COXXECTiXG    1UGHT   LltfES.          6'1 


Then  C'  will  be  pet  on  AC  prolonged  and  a  chain  from  B',  D' 
will  be  set  on  AD  prolonged  and  a  chain  from  C',  etc. 

To  prove  that  CC'  is  less  than  BB'.  Draw  GH  equal  and  paral- 
lel to  BE'.  Then  BB'CII  is  a  parallelogram,  and  B'H  —  BC  — 
B'C'  =  a  chain.  Hence  C'  is  on  the  arc  whose  center  is  B'  and 
radius  B'H.  With  C  as  center  and  radius  Ctl  =  e  describe  the 


FIG.  25. 

arc  HK  UK  lies  outside  of  the  arc  HG' ,  since  the  arcs  at  //  are 
perpendicular  to  CH  and  B'H  respectively,  and  are  limited  by  AC 
prolonged.  Hence 

CC'  <  CK  =  CII  =  BB'  =  e. 

For  the  same  reason  DD'  is  less  than  CC',  and  so  on  to  the  end 
of  the  i'U)i,  that  is,  to  the  last  stake  set  with  the  instrument  at  A. 

Suppose  the  instrument  moved  to  D'. 

Since,  in  setting  E'  from  D',  we  turn  off  fromD'A  the  same  angle 
that  we  would  turn  off  from  DA  in  order  to  set  E,  we  have  D  E' 
equal  and  parallel  to  DE,  and  therefore  EE'  is  equal  and  parallel 
to  DD' .  For  the  same  reason  FF'  is  equal  and  parallel  to  EE' 
or  DD' .  S  >  on  to  the  end  of  the  curve. 

2°.  Suppose  an  error  to  occur  in  some  other  chord  than  the  first — 
in  the  second  chord,  for  example.  B  is  supposed,  therefore,  to  be 
set  correctly.  Suppose  the  next  stake  set  at  C'  instead  of  at  C, 


FIELD-MANUAL   FOR   ENGINE 


Let  BCr  =  c  +  e.     Now  BC  -f-  CC'  >  BC',  or 

c  +  <7C"  >  c  +  e.        .-.   (7(7'  >  e. 

From  C"  the  errors  will  follow  the  same  law  as  in  the  former 
case. 

With  reference  to  these  two  cases  we  remark  that  if  the  error 
occurs  in  the  chord  adjacent  to  the  instrument,  the  errors  in  the 
stations  will  decrease  slightly  to  the  end  of  the  run,  and  from 
that  point  remain  constant  to  the  end  of  the  curve  ;  but  if  the 
error  occurs  in  a  chord  not  adjacent  to  the  instrument,  the  error 
in  the  station,  at  the  end  of  that  chord,  is  slightly  greater  than 
the  error  in  the  chord,  though  the  errors  slightly  decrease  from 
this  point  to  the  end  of  the  run,  and  remain  constant  from  the 
end  of  the  run  to  the  end  of  the  curve.  In  all  cases  the  error  at 
the  end  of  the  curve  may  be  regarded  as  equal  to  the  error  in  the 
chord,  whether  adjacent  to  the  instrument  or  not. 

3°.  Suppose  the  chain  is 
in  error,  in  which  case  all 
the  chords  will  be  in  error. 
Let  the  curve  AB  (R  = 
AO  =  radius,  and  D  —  de- 
gree) be  run  with  a  chain 
100  feet  long,  and  the  curve 
AB'(R'  =  AO'  =  radius) 
with  a  chain  100  -f-  e  feet 
long.  The  number  of 

V 

chords  is  equal  to  —  =   n. 

Then  for  the  difference  in 
length  of  the  curves  we 
have  AB'  —  (100  +  e)n. 
AB  =  ICOn,  and  therefore 


AB'  -  AB  =  en. 


(34) 


For  the  chord  we  find,  since    R'  —  lit— 


AB'  = 
and  therefore 


AB'  -  AB  =  BB'  = 


',     AB  =  21i 

ZRes'm  ^V 
~100        ' 


SIMPLE   CURVES   CONNECTING    RIGHT   LINES. 


But 


BB' 

360tfsin^y        114.60esin"-|-y 

(35) 

TtU                           D 

Also 

11'  =  (100  +  < 

<  180 

180.         57.3, 

(36) 

We  also  have 

TV  7»7. 

-BB'~'m  *  V       114-6^(sin*F)2 

(37) 

ul 

TV, 

57.3^  sin   y 
—   7?  7?  '  cos  i  y  —  •. 

(38) 

Example.— Let  e  =  .02,    T7  =  60°,  and  D  =  6°. 

The  error  of  the  curve  =  .02  X  — TT  =  -200. 

o 

114  6  X  1  X    02 
The  error  of  the  long  chord  =  —  —  =  .191. 

Also  11'  -  R  -  -   '—£—    ~  =  -19!  • 

114.6  X  .02  X 


([ 


=  .0955, 


B.  Suppose  that  the  first  stake  is  set  at  B',  Fig.  27,  instead  of 
at  B,  the  error  in  the  angle  being 
BAB'  =  A. 

Then  it  is  evident  that  Cf, 
D',  etc.,  will  be  set  at  the  same 
angular  distance  from  the  chords  p, 

AC,  AD,  etc.,  as  B'  is  set  from 
AB.  Bf,  C',  etc.,  are  on  the 
curve,  having  0'  as  center  and 
AO'  =  AO  as  radius.  OAO' 
=  BAB'. 

To  find  the  error  at  the  end  of 
the  curve. 

Suppose  .Z£the  end  of  the  true 
curve. 


C4  FIELD-MANUAL    FOll    ENGINEERS. 

Now  AE'  =  AE,  and  the  angle  EAE'  =  the  angle  BAB'. 
Hence        JE7i"  =  2^1#  sin  \EAE'  =  2AE  sin 


If  the  error  in  the  angle  is  corrected  at  the  end  of  a  run,  say  at 
IS',  then,  for  reasons  given  above,  the  error  in  the  position  of  the 
stations  is  constant  from  Er  onward  to  the  end  of  the  curve.  If 
the  error  is  corrected  during  a  run,  say  just  before  D'  is  set, 
then  that  station  will  be  set  at  Di  on  AD  and  100  feet  from  C'. 
Similarly  Ei  will  be  set  on  JJ^and  100  feet  from  2)lf 

It  may  be  shown  that  EEi  is  less  than  DDl  precisely  as  it  was 
shown  that  CO'  is  less  than  BB  '  in  Fig.  25.  Hence  in  this  case 
the  error  will  decrease  to  the  end  of  the  run,  and  remain  constant 
from  that  point  to  the  end  of  the  curve. 

II.     Curves  Laid  Out  by  Tangent  Offsets. 

If  in  Fig.  22  the  tangent  at  B  is  swung  through  an  angle  A,  say, 
then  all  stations  following  B  will  be  misplaced,  the  error  increas- 
ing to  the  end  of  the  "  run  "  in  the  manner  shown  in  Fig.  27.  If 
a  new  tangent  is  drawn  before  reaching  the  end  of  the  curve,  as 
at  E',  Fig.  27,  it  is  easy  to  see  that  such  tangent  would  make  an 
angle  A  with  the  tangent  to  the  true  curve  at  E,  and  that  the 
error,  therefore,  would  go  on  from  E'  forward  precisely  as  from 
B  to  E',  and  hence  the  error  would  increase  regularly  from  B  to 
the  end  of  the  curve. 

III.     Curves  Laid  Out  by  Ordmates  from  a  Long  Chord, 

In  this  case,  if  the  end  C  of  the  chord  CK,  Fig.  21,  is  misplaced, 
then  all  stations  from  D  to  H  inclusive  will  be  misplaced  in  pro- 
portion to  their  distances  from  K\  the  greatest  displacement  being 
less  than  that  of  G.  If  K  is  also  misplaced,  the  same  stations 
will  likewise  share  that  error,  in  proportion  to  their  distances 
from  C. 

IV.   Curves  Laid  Out  by  Chord  Offsets.     (Fig.  22.) 

If  in  this  method  any  station,  as  B,  is  set  at  one  side  of  its  true 
position,  then  all  subsequent  stations  will  be  in  error  in  the  same 
direction  ;  the  errors  increasing  regularly  to  the  end  of  the  curve, 
in  the  manner  shown  in  Fig.  27. 

If,  owing  to  an  error  in  some  chord,  some  station  is  set  forward 
or  backward  from  its  true  position,  then  all  subsequent  stations 
will  be  in  error  the  same  amount  and  in  the  same  direction. 


SIMPLE    CU-RVES    CONNECTING    RIGHT    LINES. 


65 


V.  Curves  Located  by  Middle  Ordinntes. 

In  Fig.  23  suppose  G'to  be  placed  a  distance  a  to  the  right,  say, 
(that  is,  along  CO,}  of  its  true  position.  Then  c  will  be  a  distance 
ft  to  the  right  ;  D  and  d  will  be  2a  to  the  right  ;  ^and  e  will  be 
'*i  to  the  right  of  their  true  positions,  etc. 

If  C  is  correct,  but  c  a  distance  a  too  far  to  the  right,  then  D 
•.nd  d  will  be  2a  to  the  right,  E  nud  e  will  be  4«  to  the  right,  F 
ind/  will  be  $a  to  the  right  of  their  true  positions,  etc. 

PROBLEMS  IN  SIMPLE  CURVES. 

I.  Given  the  tangent  distance  AB  =  d  and  the  tangent  offset 
BD  =  t,  to  find  the  radius  of  a  curve  that  will  pass  through  D 
ind  be  tangent  to  AB  at  A. 

Let  AD  =  c.     We  have 


From  this  we  have 


1  -  d\    .     .     (39) 


FIG.  28. 


and  d  =  4/t(2JZ  -  t).     .     . 

Second  Solution. — Let  AOD  =  A  ;  then 

BAD  =  ADC  =  %A. 
Now  the  triangle  ABD  gives 

d-  =  cot  $A.     .     . 


(40) 


Also 
and 


-  =  vers  ^4, 
—  =  cosec  A. 

a 


(41) 
(42) 
(43) 


66  FIELD-MANUAL    FOR    ENGINEERS. 

If  d  and  t  are  given,.  find  A  from  (41),  then  R  from  (42)  or  (43) 
If  t  and  .7?  are  given,  find  A  from  (42),  then  d  from  (41)  or  (43). 
Finally,  if  d  and  R  are  given,  find  A  from  (43),  then  t  from  (41) 
or  (42). 

Example  1.  —  Given  t  =  12  and  d  =  171.6,  to  find  It. 

tan  §A  =  p^-g  =  .06993. 
.;.     \A  =  4°,     and     A  =  8°. 

Then  R  =  _  -  _!_    -  --^_  =  1233.3. 

vers  .4         .00973 

Example  2.~  Given  72  =  1233.3  and  d  =  171.6,  to  find  t. 
cosec  A  =  -^8j|-  =  7.187.     .-.   4  =  8°. 

Then  t  =  1233.3  X  .00973  =  12. 

This  problem  is  useful  in  finding  points  on  a  curve  beyond  an 
obstacle. 

II.   To  find  the  distance  to  a  curve  in  a  given  direction  from  a 
given  point  on  a  tangent. 
We  have 

AO  =  11,     AS  =  d,     and     ABP  =  B. 

tan  ABO  =  ?  .  PBO  =  ABO  -  ABP; 
d 

sin  OPI  =  sin  OPB  =  ~  X  sin 
ji 

sin 


••  sin  ABO' 
Then 

1  sm(OP2-PBO) 

FIG.  29.  —faTPBO 

This  problem  furnishes  a  general  method  of  finding  any  desired 
point  on  a  curve  when  obstacles  preclude  the  usual  methods. 
(See  Problem  12.) 

III.  Having  run  the  curve  AD,  radius  AO  =  It,  Fig.  28,  to 
find  the  radus  R'  of  a  curve  that  will  puss  through  D',  given  by 
angle  BDD'  =  D,  and  DD'  =  E. 


SIMPLE    CURVES    CONNECTING    RIGHT   LINES.          67 

Let  AB  =  d,     and     BD  =  t. 
Draw  D'H  parallel  to  AB.     Now 

HD'  =  E  sin  D,     and    1W  =  E  cos  D. 
AB'  =  d+  #sin  D  =  dlt     and     B'D'  =  t  -  E  cos  D  =  ti. 


It  will  be  observed  that  when  AB'  <  AB,  E  sin  Z>  must  be 
subtracted  from  d  to  give  d* ;  and  that  when  B'D'  =  BH  >  BD, 
E  cos  D  must  be  added  to  t  to  give  £,. 

IV.  Having  run  a  curve  of  radius  R  and  tangent  T,  to  find  the 
new  tangent  T'  corresponding  to  a  new  radius  R' ',  or  to  find  a  new 
radius  h'  corresponding  to  a  new  tangent  T' ',  the  central  angle  re- 
maining constant. 

Eq.  (15)  gives 

T'  =  R'  tan  \V\     T  =  R  tan  % V\ 
.'.   T'  -  T—  (11'  -  R)  tan  *  F,     .     .     .     .     (45) 
or  72'  —  R  =  (T' —  T)  cot  \V.    '..'..     (46) 

Similarly,  from  eq.  (17),  we  get 

C'  —  C  =  2(72'  —  R)  sin  ^F.      ....     (47) 

These  equations  are  of  advantage  for  computing  the  change  in 
one  element,  T'  —  T  for  example,  from  the  change  in  another, 
R'  —  R  for  example,  when  the  given  change  is  small,  or  is  an 
aliquot  part  of  the  element  changed. 

Example  1. — Having  run  the  curve  of  radius  72  =  1910,  and  the 
central  angle  V=  46°  12',  and  tangent  distance  T  =  R  tan  \  V 
=  1910  X- 42654  =  814.7,  to  find  T'  when  72  is  made  equal  to  1900. 

Eq.  (45)  gives 

=  814.7  -  .42654  X  10  =  814.7  -  4.27  =  810.4. 
Example   2.—  Given  F  =  52°  04',    72  =  5730,   and    T  =  .48845 
X  .5730  =  2798.8,  to  find  72'  corresponding  to  T'  —  T  +  •-- . 
We  have 

R'  =  R  +  ^  =  2798.8  +  233.2  =  8032.0. 


68 


FIELD-MANUAL    FOR    ENGINEERS. 


V.  (riven  a  curve  joining'  two  tangents,  to  change  the  position 
of  the  P.  C.  so  that  with  the  same  radius  the  curve  may  end  in 

a  given  parallel  tangent. 

Let  ^47?  be  the  given  curve, 
and  IV  F'the  parallel  tangent. 
W' —a  shows  the  distance 
and  direction  that  all  points  of 
the  curve  are  moved.  The 
curve  will  therefore  begin  at 
A '  and  end  at  Bf ;  A  A  and  BB', 
as  well  as  00',  being'  equal 
and  parallel  to  YV.  It  is 
not  necessary  to  run  the  tan- 
gent B'  V  in  order  to  find  the 
distance  VV.  To  find  this 


FIG.  30. 


distance  run  a  line,  such  as  BB',  parallel  to  A  V  from  any  point 
on  BV  to  meet  B'V.     Then  make  A  A'  =  BB'. 

If  the  perpendicular  offset  Bh  —  7t  is  measured,  we  have 


AA  =  BB'  = 


V  being  the  vertex  angle. 

It  B'V  were  on  the  other  side  of  .BFfrom  that  shown,  the 
new  tangent  point  A'  would  fall  on  the  opposite  side  of  A  from 
that  shown  in  the  figure. 

If  the  new  curve  is  required  to  end  at  a  given  point  on  B'V, 


we  .have,   then,   the 
new  tangents, 


length   of   the 


A'V  =  B'V  =  T', 

which  gives  the  position  of  A  (and 
B'),  and  the  corresponding  radius, 

It'  =  7"  cot  IF, 
or,  by  (46), 

R'  =,  R  +  (T'  -  T)  cot  *F. 

FIG.  31. 

VI.  Given  a  curve  AB  joining  two  tangents,  to  find  the  radius 
of  a  curve  that  from  the  same  P,  C.  will  end  in  a  given  parallel 
tangent. 


SIMPLE    CURVES    CONNECTING    RIGHT    LINES.  09 

Let^F=£F  =  T,  AV'  =  B'V'  =  T't  AO  =  R,  A0f  =  R'. 

We  liave,  from  the  figure, 

R'       T'  7" 

R=-T>     °r     ll     =ET- 

Also,  from  eq.  (46), 

R'  =  R  +  (Tf  -  T)  cot£F. 

Or,  prolong  AB  to  B'  and    measure  BB'.       Let  ^45  =  c,  and 
A#'  =  c',  BB'  —  c'  -  c.      Then,  from  the  figure  or  from  (17), 


If  the  parallel  tangent  is  defined  by  a  perpendicular  offset,  as 
B'p  —  Ji,  draw  BG  parallel  to  AO.  Then 

Cp  =  BCcos  BCp  =  (R1  -  R)  cos  V. 

.-.  CB'  =  (R'-R)  =  (R'-K)coz  V+U,     or     (1?  -£)(!-  cos  V)=7i, 
or          (R'  -  R)  versin  V  =  h,     or     R'  =  R  +  -      A 

The  quantity  added  to  IMn  the  above  equations  must  be  sub- 
tracted from  it  to  find  R'  in  the  case  in  which  V  falls  between  A 
and  V,  that  is,  when  T'  is  less  than  T7.  • 

Example  1.  —  V  --  78°,  R  =  954.9.  T.7  may  be  computed  or  found 
by  Table  VII  to  be  773.3. 

Let  VV  =  20  feet.     Then 

R'  =R+  (T'  -  T)  cot  £F=  954.9  +  20  X  1.2349  =  979.6. 

Example  2.—R  —  1909.9,  F=  46°  38'.  T  may  be  computed  or 
found  by  Table  VII  to  be  823.2. 

It  is  desirable  to  move  the  vertex  from  Fto  V  about  100  feet. 
Find  the  new  radius  R'. 

823.2  -*  8  =  102.9; 
1909.9  H-  8  =  238.7, 

Hence  A  V  -    823.2  +  102.90  =  926.1, 

and  the  new  radius 

AO'  =  If  _  1909.9  +  238,7    =  2148.6. 


70 


FIELD-MANUAL   FOR   ENGINEERS. 


VII.  Given  a  curve  joining  two  tangents,  to  find  the  new  tan- 
gent points,  corresponding  to  the 
same  radius,  after  eacli  tangent  Las 
been  moved  any  distance  in  the  di- 
rection of  the  other. 

Let  A  Fand  .Z?Fbe  the  given  and 
A' V  and  B'V  the  required  tan- 
gents. Let  Hbe  at  the  intersection 
of  AV  &ul  B'V.  Let  VH  =  a, 
VII  =  b,  and  VV  =  c. 

We  observe  that 


B 


VHV  =  180°  -  F. 
.'.  sin  VII V  —  sin  V, 

FIG.  32.  and         cos  VHV  =  —  cos  F. 

Chap.  Ill,  formula  No.  (8),  gives 

sin  F  sin  F 


HV 


r-f  -j-  COS    V 


cos  V 


Then 


VV  = 


b  sin  V 


sinhVV 


We  observe  that  the  directions  of  F//and  HV  are  the  same 
as  that  in  which  the  tangents  B  V  and  .4  Fare  moved,  and  there- 
fore there  can  be  no  ambiguity  about  the  direction  of  these  lines 
or  of  FF',  which  is  the  line  joining  F  and  V. 

Since  R  and  F  are  not  changed,  it  is  evident  that  all  parts  of 
the  curve  are  moved  in  the  direction  VV  and  a  distance  equal  to 
FF'. 

Hence  make  the  angle  VAA'  =  HVV,  and  A  A'  =  VV. 

The  curve  will  begin  at  A'  and  end  at  B'  ,  BB'  as  well  as  00r 
being  equal  and  parallel  to  W  . 

If  the  distances  the  tangents  are  moved  are  given  by  perpendic- 
ular offsets  Vh'  =  hf,  and  Vh  =  h,  the  triangles  Vllh  and 
V  '  Hh'  are  similar  and  give 


HV 
HV 


sin  VHV 


HV 
HV' 


or 


SIMPLE   CURVES   CONNECTING    RIGHT   LINES. 

sin  V 
~  h 


71 


Now  the  triangle  hVV '  gives 

vv-    -7^- 

smHW 

VV  and  Vh  are  on  the  same  side  of  B  V;  also,   VV,  and  Vh> 
are  on  the  same  side  of  A  V. 

VIII.  Given  a  curve  AB  joining  two  tangents  AV  and  BV,  to 
change  the  curve  so  as  to  end  at  the 

same  point  as  before,  bat  in  a  tan-    A.  A'  V        V'     D 

gent    inclined  at  a  given   angle  A 
with  the  original  tangent. 

Let  A  V-  BV  =  T,  and  the  new 
tangents  A'V  =  BV  =  T'. 

Let  AO  =  /?,     and      A'O'  =  II'. 


Draw  BD  =  p  perpendicular,  and 
BMN  parallel  to  A  V. 
Let 

V'=BV'D=BVD+VBV'=V+A. 

Now     An  =  It  versin  V, 
and         A'm  —  11'  versin  V. 
But        A'm  =  An', 


B 


FIG.  33. 

. '.  R  versin  V  =  R'  versiu  V, 
R  versin  V 


R'  = 


versin  V 


With  this  value  of  R'  run  the  curve  back  from  B  through  the 
angle  V,  and  it  will  end  at  A',  tangent  to  A  V\  A'  V  being  equal 
to  BV. 

If  the-length  of  the  new  tangent  is  desired,  we  have,  from  the 
triangle  VBV, 


Then 


sin  V 
T' 


sin  V  ~  sin  V 


P 


tan 


sin  V  tan  \V  ~  vers  V 


72  FIELD-MANUAL    FOR    ENGINEERS. 

If  we  wish  to  run  the  curve  from  A,  we  have 

D  V  =  p  cot  V,     and     D  V  =  p  cot  V. 

.-.   VV  =p(cot  V-  cot  V), 
and         AA'  =  AV+  VV  -  A'V  =  VV  -\-T-T'. 

When  V  <  90°,  T  increases  as  V  decreases,  and  vice  versa; 
and  when  V>  90°,  I7  and  V  increase  and  decrease  together.  In 
all  cases  R  increases  as  V decreases,  and  vice  versa. 

IX.  Given    a    curve,    radius    AO  =  R,   joining  the   tangents 

A  V  and  BV,  to  find  the  radius 
AO'  =  R'  of  a  new  curve  start- 
ing from  A  when  the  forward 
tangent  VB'  takes  a  new  direc- 
tion from  the  vertex. 
"We  have 


and 


VA  =  R  tan  \Vt 
R'  =  VA  cot£F. 


tan 


FIG.  34. 


tan  |  V 


X.  Given  a  curve  AB,  radius  AO  —  R,  joining  the  tangents 
A  Fand  VB,  to  find  the  change 
in  the  P.C.,  the  radius  remain- 
ing the  same,  when  the  forward 
tangent  takes  a  new  direction 
from  the  vertex. 

We  have 

VA  =  5  tan  £F; 
VA'  =  Btan^F. 


FIG-  35. 


XL     Given  the  angle  of 
tersection    V  of  two  tangents  0 
4  Fand  BV>  tP  find  the  radius 


SIMPLE   CURVES   CONNECTING    RIGHT   LINES. 


R  and  tangent  distance    T  of  u   curve  joining  the  tangents  and 
passing  tb  rough  the  point  E. 

1°.  Let  E  be  given  by  VE=l, 
and  angle  EVO  =  A. 

Let    VEO  =  E,     VOE  =  0, 
and    AO  =  21,   AOV  =  $V. 

No\v 

i  v-  A0  -  E0  -  ^A. 

-Jd~'VO~ 


sin 


sin  E  — 


sin  A 


cos  \  V 
This  gives  E.     Then 

0  =  180°  -  (A  4  E). 
Moreover, 

EO     _  R  _   sin  A 

~EV  ~~  T  " ~    sin  0 


sin  A 

or      7?  —  I  -. — — 
sin  0 


Finally,  T= 

2°.   If  E  is  given  by   VZZ"  and  HE  perpendicular  to  each  other, 
EH 


then 


tan  E  VII  = 


VII' 


EVF  =  FVH  -  EVH  =  90°  -  £F -  EVH; 

VII 

~~  cos  EVH ' 

With  these  values  proceed  as  above. 

3°.  Let  Ebe  given  by  VD  —  a,  IXfiJ  =  ft,  the  angle  VDF  being 
rial  to  FO^l  =  4F.     Produce  i>^to  2^ and  (7. 


Let  DF  =  c.    Then  c  —  a  cos  ^F,  and  AD  =  \/DE  X  DO. 
But    DO  =  DF+FG  =  DF -\-  EF=2DF-  DE=2c-b. 
~1>),     and     VA=  VD  +  DA  =  T. 


.-,  AD= 
Now   It=  Tcot  \V, 
or         HE  =  DE  sin 


=  b  sin  i  F. 


74  FIELD-MANUAL   FOR  ENGINEERS. 

D1I  =  b  cos  £  V. 
Then  VH=  VD  —  DH  =  a  -  b  cos  \V. 

With  these  values  of  F/iTand  HE  proceed  as  above. 

XII.  Given  a  tangent  and  curve  (Fig.  36),  to  find  the  distance 
from  a  given  point  on  the  tangent  to  the  curve  in  a  given  direc- 
tion. 

Let  V  be  the  point,  and  suppose  the  direction  defined  by  the 
angle  EVA  =  B.  Let  AV  =  T.  We  have 

T 
taniF=-. 

Then        EVF  =  D  VF  -  EVA  =  (90°  -$V)-B  =  A,  say. 
Now  equation  under  Problem  XI  gives 

sin  A 

sin  E  =  --  T^pf. 
cos^F 

Then  0  =  180°  -(A  +  E), 


or, 


^,     and    EVF  =  FVD  -  EVD  =  FDV  -  B  =  A. 


Then  find  sin  J^,  then  0  and  7  as  before. 
Example.—  R  =  954.93,  T7  =  A  V  =  350,  ^L  VE=  40°.     We  have 


tan  AVO  =         -  2.72837;     .'.  J[FO  =  69°  52', 
and  AOV=±V=  20°  8',    and    #FF  =  29°  52'  =  F. 


0  -  180°  -  177°  50'  =  2°  10'; 

_  954.93  X  -03781  _  70  4 
.49798" 


SIMPLE    CURVES    CONNECTING    RIGHT    LIKES. 


75 


XIII.  To  locate  a  tangent  to  a  curve  of  given  radius  R  from  a 
given  point  V.     (Fig.  37.)   v  c 

1 .  If  the  curve  is  marked 
by  stakes  visible  from  the 
given  point,  a  tangent  can 
be  sighted  in  at  once. 

2.  If    the    curve    is   not 
visible  from  the  point,  run 

a   trial   tangent    VB,    and  FIG.  3T. 

measure  VB  —  A  arid  the  angle  VBO  =  B. 
Chapter  III,  formula  (8),  gives 

sin  B 


tan^FO  = 


Then          OV=OB. 
Lastly, 


sin  B 


—  —  cos  B 


and     sin  EVO  = 


sin  B  VO" 
BVE  =  EVO  -  BVO. 


EO 

or 


sin  A 


-  —  cos  A 


This  gives  the  angle  to  be  laid  off  from 
the  trial  tangent  to  give  the  true  tan- 
gent AE. 

XIV.  Given  two  curves  AB  and  AC, 
radii  AO  =  R,  AOi  =  R\,  subtending 
the  central  angles  AOB  =  0  and 
AO^C  =  0,,  to  find  the  length  of  the 
line  EC.  (Fig.  38.) 

Let  A,  B,  and  G  represent  the  angles 
of  the  triangle  ABC,  and  «,  b,  and  c  the 
sides  opposite.  We  have 

c  =  2R  sin  \0,     and    &  =  2R}  sin  |0,. 
Also, 

BAO  =  90°  -  \0,     CAOv  =  90°  -  £0, 
.-.  BAC=  A  =  1(0  -  0i). 

-,    (See  Chap.  Ill,  formula  (8).) 


and  then 


, 

a  =  b 


sin  A 
-  —  ~. 
sin  B 


FIELD-MANUAL   Foil    EXGIXKEKS. 


If  the  curves  are  run  through  an  integral  number  of  stations, 
Tables  IV  and  V  give  at  once  the  tangent  distances  AK  and  AH 
and  the  tangent  offsets  BK  and  CH.     Then,  drawing  CD  parallel 
to  A  //to  meet  BK'ni  D,  we  have 
CD  =  AH  -  AK  =  d,  say,     and     BD  =  BK  -  Ctf=  t,  say. 

Then  BC  =  f < 


and 


BC  = 


BD 


OBSTACLES  IN  SURVEYING. 

It  is  often  necessary  to  draw  lines  parallel  and  perpendicular  to 

other  lines.     Hence  the  follow- 
ing problems  : 

I.   To  erect  a  perpendicular  at 
any  point  cf  a  line.     (Fig.  39.) 

1.  Let  J.  be  the  point,  and  EC 
the  line.     Make  AE  =  AC,  and 
with  B  -and   C  as   centers   and 
any  radius  greater  than  AE  de- 
scribe arcs  intersecting  at  D  or 
at  E,  or  (with  a  different  radium) 
at  F.     Any  two  of  the  points  A, 
D,  E,  and  F  determine  the  per- 
pendicular required.  "! 

2.  Fix  any  two  points  of  the 
chain  at  B  and  at  C.     Take  hold 
of  the  point  midway  between  B 
and  G  and  stretch  the  chain,  the 

middle  point  being  at  D.  AD  is  the  perpendicular  required. 
We  may  find,  similarly,  other  points  E,  F,  etc.  Any  two  of  these 
points  A,  D,  E,  F,  etc.,  determine  the  perpendicular  required. 

3.  Let  C  be  the  point.     Take  any  point  D  as  a  center,  and  with 
a  radius  DC  describe  an  arc  BC.   Prolong  BD,  making  DH  —  BD. 
CIlis  the  perpendicular  required.     For  DA  (A  being  at  the  mid- 
dle of  BC)  is  perpendicular  to  BC,  and,  by  construction,  CH  is 
parallel  to  AD. 

4.  A  right  angle  may  be  obtained  by  laying  off  on  the  ground 
the  three  sides  of  any  of  the  triangles  represented  in  the  following 
table,  or  any  equimultiples  of  these   sides,   making  one  of   the 


FIG.  39. 


8IMPLK   CURVES   CONNECTING    RIGHT    LINES. 


77 


sides  adjacent  to  the  right  angle  (a  or  b)  coincide  with  the  line. 
Let  c  =  the  hypothenuse. 


No. 

u. 

b. 

c. 

No. 

a. 

b. 

C. 

1 

3 

4 

5 

6 

20 

SI 

29 

2 

5 

12 

13 

7 

12 

33 

37 

3 

8 

15 

17 

8 

9 

40 

41 

4 

24 

25 

9 

11 

60 

61 

5 

10 

24 

26 

10 

13 

84 

85 

B 


Thus,  in  Fig.  40,  using  70  links  of  the  chain,  hold  the  first 
end,  also  the  end  of  the  70th  link  of  the  chain,  at  A,  the  end  of 
the  21st  link  at  B,  and  the  end  of 
the  50th  link  at  C. 

If  in  the  three  expressions 
ra2  —  ir,  2mn,  and  m2  -j-  ifi  we 
assign  to  m  and  n  any  values  at 
pleasure,  in  being  greater  than  n, 
we  will  have  sets  of  numbers 
representing  the  sides  of  right- 
angled  triangles.  In  that  way 
the  above  numbers  were  found. 
Equimultiples  of  any  set  of  the 
above  numbers  will  represent  the 
sides  of  a  right-angled  triangle. 

II.  To  let  fall  a  perpendicular  from  a  given  point  to  a  given 
line. — Let //in  Fig.  39  be  the  point.  Measure  any  line  HB  to 
the  given  line.  At  the  middle  of  ///>  take  D  as  a  center,  and 
with  a  radius  DB  describe  an  arc  BC.  HO  is  the  required  per- 
pendicular. For  continuing  the  arc  to  II,  we  see  that  the  angle 
BCIIis  inscribed  in  a  semicircle. 

III.  To  let  fall  a  perpendicular  to  a 
line  from  an  inaccessible  point. — Let 
BC  (Fig.  41)  be  the  line,  and  P  the 
point.  Let  p  represent  the  perpen- 
dicular PK.  Then 

BK  —  p  cot  B,   and    CK  =  p  cot  C. 

BK  _  cot  B 
•''   CK  ~ 


21 
Fm.  40. 


FIG.  41. 


and 


BK 


cot  C' 

cot  B 
cot  B  -j-  cot  C ' 


78  FIKLD-MAXUAL    FOR    EXGTXEKRS. 

Since  BK  -\-  CK  =  BC,  we  have 
BK  =  BC 


cot  B  -f  cot  0  * 

Tills  gives  the  foot  of  the  perpendicular  K.  If  BC  is  taken  equal 
to  100  or  some  small  multiple  of  100,  BK  is  very  easily  found. 

This  problem  is  particularly  useful  in  locating  important  ob- 
jects, such  as  mills,  warehouses,  bridges,  etc.,  on  one  side  or  the 
other  of  a  railway  survey. 

In  this  case  B  and  C  represent  stations  or  points  on  the  survey, 
and  the  angles  at  B  and  G  can  be  measured  and  recorded  while 
the  instrument  is  set  at  B  and  at  C.  The  simple  divi-ion  required 
to  find  the  position  of  .STcan  be  made  at  any  time.  Of  course  the 
point  Pis  located  graphically  by  drawing  .Z?Pand  CP. 

IV.  To  prolong  a  line  AB  (Fig.  42)  past  an  obstacle  and  to 
measure  its  length. — This  is  easily  done  by  perpendicular  offsets, 
a  method  to  >  familiar  to  need  description,  but  not  the  best  way. 


B 


FIG.  42. 

Or,  measure  BC  in  any  convenient  direction,  and  at  C  deflect  any 
angle  FCD.  Draw  BD  and  the  perpendicular  CH.  The  angle 
CDS  =  FCD  -  CBD.  Hence 


sin  FCD 
also  <P-L>  =  ^^  sin  BBC' 

If  the  angle  BCD  is  made  equal  to  90°,  then 

CD  =  BC  tan  CBD, 

and  BD  =  BC  -f-  cos  CBD. 


SIMPLE   CURVES    CONNECTING    RIGHT    LINES. 


If  the  angle  FCT)  is  made  equal  to  2CBD,  tlien 
CDS  =  FCD  -  CBD  =  CBD. 
Hence         CD  -  BC,     and     BD  —  2BH  =  2BC  cos  CBH. 

OH  is  tlie  departure  of  the  line  BC,  or  of  DC,  from  the  line 
ABD.  It  is  also  the  approach  of  the  line  CB,  or  of  CD,  to  ABD. 

If  necessary  more  than  one  course  may  be  run  away  from  the 
main  line  ABD,  and  more  than  one  in  returning  to  it. 

To  recover  the  main  line  it  is  only  necessary  to  make  the  sum 
of  the  approaches  equal  to  the  sum  of  the  departures. 

The  distance  measured  on  the  main  line  is  obtained  as  above. 

V.  Obstacles  to  Measurement. — Methods  have  been  pointed  out 
in  connection  with  Fig.  42  for  rinding  the  length  of  obstructed  lines 
when  the  ends  are  accessible.  When  inaccessible  the  following- 
problems  apply. 

A.  When  one  end  of  the  line  is  inaccessible.     (Fig.  43.) 

1.  Let  AB  be  the  line  to  be  measured,  across  a  river  for  ex- 
ample. Measure  AC  in  any 


convenient  direction,  and  the 
angles  at  A  and  C.     Then 

AC  sin  C 

AB  =  — 


B 


sin  B 


=  AC-. 


sin  C 


sin  (A  +  C)' 

2.  If  the  angle  ACS  is 
made  equal  to  half  of  DAC, 
then 

OB  A  =  CAD  -  BCA 


.'.   AB  -  AC.  FIG.  43. 

3.  Or,  in  Fig.  43,  make  the  angle  BAG  =  90°.     Then 

AB  =  AC  tan  ACB. 

If,  in  this,  AC  =  100,  or  some  simple  multiple  of  100,  which  is 
usually  easy  to  effect,  the  formula  requires  no  computation 
whatever. 

4.  If  ACB  in  Fig.  44  is  made  equal  to  45°,  AB  —  AC. 


80 


FIELD-MAXUAL    FOR   EXGIXEERS. 


5.  If  at  G  we  make  the  angles  ACB  and  ACD  equal,  we  liavo 
AB  =  AD. 

When  the  river  or  other  obstruction  occurs  on  a  continuous 
survey,  as  a  railway  survey,  AD  is  a  measured  line,  and  this 
method  gives  AB  =  AD  without  any  computation  whatever. 


FIG.  45. 


6.  In  Fig.  45,  AB  being  the  distance  required,  run  and  measure 
any  line  AC  and  measure  the  angle  BAG  —  A.     Make 


Then 


ACB  =  90°  -  A  =  C. 
AB  =  AC  sin  C. 


B.  When  both  ends  of  the  line  are  inaccessible.     (Fig.  46.) 
Let  AB  be*  the  line  to  be  meas- 


ured. Find  the  distances  from  the    A- 
point  G  to  each  end  of  the  line  A 
and  B  by  preceding  methods,  and 
measure  the  angle  G.     Then 

sin  G 


tan  A  = 


AC 


(see  Chapter  III,  formula  (8).) 


C 
FIG.  46. 


. 

sin  A 


C.  To  erect,  at  a  given  point  A  (Fig.  47),  a  line  ylZf  perpendicu- 


SIMPLE    CURVES    CONNECTING    RIGHT    LINES. 


81 


\ 


lar  to  an  inaccessible  line 
BC,  and  to  draw  a  parallel 
AH  to  the  same  line. — Find 
.47?  =  c  and  AC  —  b  by  pre- 
ceding methods.  Then 

pin  A 
tan  B  =  -        . 

- —  cos  A 
I) 

Now  draw  A  K,  making 
BA  K  =  90°  -  B. 

A  K  will  be  the  required  per- 
pendicular, and  AH,  making  BAH  —  B,  will    be    the    required 
parallel. 

D.   To  find  the  length  and  relative  position  of  an  inaccessible  lino, 
AB,  from  an  accessible  line,  CD,  separated 
n  from  the  former  by  an  inaccessible  space.    - 
Measure  CD  and  the  angles  at  C  and  D. 
Example.— Let  CD  =  4000; 

BCD  =  126°  25V;    ADC  =  47°  53 V; 
ACB=      3°  10';      ADB=    3°  01'. 
.-.  ACD  =  129°  35V;    SJ)C  =  50°  54 1'. 
Also, 

CAD  =  180°  -  ACD  —  ADC  —  2°  31'; 
CBD  =  180°  -  BCD  -  BDC  =  2°  40'. 
Hence 

40=  400o5!2*™a.'  =  67«81.»; 


BC  =  4000 
tan  CAB  = 


sin  2°  40' 
sin  C 


=  66728.8; 


'AC 


FIG,  48, 


Finally, 


CAB  =  75°  28V. 
^sin    3°  10' 


-,  -  3807.8. 


82  FIELD-MANUAL    FOH   ENGINEERS. 

If  AD  is  accessible,  it  can  be  measured  as  a  check  on  the  com- 
putation. Such  is  the  case  when  it  is  a  tangent  of  a  railway  sur- 
vey adjacent  to  the  inaccessible  space. 

The  data  of  this  example  are  taken  from  an  actual  night  survey 
across  an  inaccessible  sea-marsh. 

Rockets  were  thrown  and  lights  then  exhibited  at  A.  and  B. 
which  were  observed  with  transits  from  the  tops  of  towers  at  C 
and  D. 

The  computed  and  the  measured  length  of  AB  agreed  within  a 
few  inches. 

The  best  method  of  making  a  preliminary  railway  survey 
through  a  wooded  region  is  by  a  suitable  adaptation  of  the  method 
of  traversing,  which  we  will  now  explain. 

So  far  as  known  to  the  author,  this  was  first  devised  by  him  in 
1869,  and  used  for  him  by  his  assistant,  Prof.  J.  B.  Davis  (now  of 
Michigan  University)  in  making  the  preliminary  surveys  of  the 
Owosso  and  Northwestern  Railway. 

Suppose  we  wish  to  run  from  A  in  the  direction  of  ABF,  which 
we  will  call  the  base  line;  and  upon  which  numerous  obstacles,  such 


FIG.  49. 

as  trees,  occur,  making  it  necessary  to  run  the  line  ABiCiDiE[F, 
called  a  traverse.  The  deflection  angles  at  Bi,  C\t  Diy  pud  E^  are 
supposed  to  be  small. 

From  Bi ,  d  ,  D\  ,  and  Ei  draw  perpendiculars  to  AB,  and  from 
Si,  C>,  and  Dl  draw  parallels  BiK,  dL,  and  D,P  to  AB  as 
shown.  Prolong  ABi  to  R,  and  B}  Ci  to  8. 

The  course  of  a  line  is  its  direction  with  reference  to  the 
base  line. 

The  departure  of  a  line  is  the  distance  that  a  point  recedes 
from  or  approaches  to  the  base  line  in  moving  from  one  end  of 
the  line  to  the  other. 

We  have  D1L  —  CiDi  sin  DiCiL.  Now  since  the  sines  of 
small  angles  vary  nearly  with  the  angles  or  with  the  number  of 
minutes  in  the  angles,  we  see  that  the  departure  of  a  line  varies 


SIMPLE   CUHVES   CONNECTING   1UGHT    LINES. 


83 


nearly  as  the  product  of  the  length  of  the  line  by  the  number  of 
minutes  in  the  course. 

The  departure  of  a  point  is  its  distance  from  the  base  line. 
Thus  the  departure  of  Z>,  =  DD,. 

The  departure  of  the  end  of  a  line,  as  B\  C\ ,  inclining  from  the 
base,  is  equal  to  the  departure  of  the  beginning  of  the  line  plus 
the  departure  of  the  line.  Thus  Cd  =  #77,  +  f,7i.  The 
departure  of  the  end  of  a  line,  as  C\D\ ,  inclining  toward  the  base, 
is  equal  to  the  departure  of  the  beginning  of  the  line  minus  the 
departure  of  the  line.  Thus  DDi  =  CC\  —  DiL.  The  departure 
of  the  end  of  the  line  D\E\  which  crosses  the  base  line  is  equal 
to  the  departure  of  the  line  minus  the  departure  of  the  beginning 
of  the  line.  Thus  EEl  =  E,P  -  DDt. 

The  record  of  the  survey  can  be  conveniently  kept,  as  shown 
in  the  following  table,  the  columns  of 'the  transit-book  serving 
the  purpose  perfectly. 


Angles  turned. 

Angles  with 
Main  Line. 

Departures. 

Stations. 

Each  Course. 

Total. 

Left. 

Right. 

Left. 

Right. 

Left. 

Right. 

Lcfr. 

Right. 

A  =      10 

10' 

10' 

11 

12 

£,=      13 

22' 

32' 

3000 

3000 

14 

15 

Ci  =  +  20 

4'2X 

10' 

7040 

10040 

16 

17 

18 

Z>!=      19 

3C' 

40' 

3800 

6240 

21 

E,=      24 

1°20' 

40 

20000 

137GO 

25 

26 

27 

F  =+44 

13760 

.00 

.00 

The  deflection  at  station  10  is  10'  R.>  and  at  station  13  it  is 
22'  R.,  etc.  The  course  from  10  to  13  is  evidently  10'  R. ;  from 
13  to  15  -f-  20  it  is  10  -f  22  =  32'  I*.;  from  15  +  20  to  19  it  ig 
42  -  32  =  10'  L.,  etc. 


84  FIELD-MANUAL    FOR   ENGINEERS. 

From  10  to  13  the  departure  is          300x10  =    3000  foot-minutes. 
"      13  "  15+20  the  departure  is  220x32  =    7040 
"      15+20  to  19    "  "          "380X10:=    3800 

'<      19  to  24  "          "500X40=20000       "         etc. 

The  aggregate  departures  are  : 

At  13  ................................  3000  R. 

At  15  +  20  ......  3000  +  7040  =  10040  R. 

At  19  ............  3000  +  7040  -  3800  =  6240  R.,  etc. 

The  distance  necessary  to  run  from  a  given  station  on  any  given 
course  to  reach  the  base  line  is  found  by  dividing  the  departure 
at  that  station  by  the  course.  Thus  from  station  24  forward  the 
course  is  40'  R.  Then  13760  -4-  40  —  344  feet,  showing  that  the 
auxiliary  line  (E^Fin  the  figure)  will  reach  the  base  line  344  feet 
beyond  station  24,  or  at  27  +  44. 

The  figure  represents  a  main  angle  at  F,  the  forward  tangent 
being  FIT,  and  which  may  be  followed  approximately  the  same  as 
AF  WAS  followed. 

Let  I  =  the  length  and  d  =  the  departure  of  any  line,  and  n  — 
the  number  of  minutes  in  the  course.  Then 

d  =  I  sin  n'  =  In  sin  1'  =  .0002909fo. 

Since  In  is  given  in  the  last  two  columns,  the  departures  in  feet 
are  found  by  multiplying  the  quantities  in  these  columns  by 
.0002909  or  .00029  nearly.  We  observe  that 

1  4-  .0002909  =  3438  nearly. 

Hence  the  departures  given  in  the  table  (in  foot-minutes) 
divided  by  3438  will  give  the  departures  in  feet. 

A  rough  approximation  for  the  purpose  of  keeping  sufficiently 
near  the  base  line  on  sideling  ground  is  generally  all  that  is 
needed.  This  being  the  case,  it  is  not  in  general  necessary  or 
advisable  to  find  the  total  departures,  except  when  it  is  desirable 
to  "  run  for  the  base  "  preparatory  to  turning  a  main  angle. 

Thus  to  find  the  departure  at   station   24.     The   sum   of  the 
product  to  the  left  is  ____  .............  3800  +  20000  =  23800 

and  the  same  to  the  right  is  ...........  3000  +    7040  =  10040 

The  difference  is  ..................................  13760  L. 

We  have  AR  =  AB,  cos 


~  4W  -  <-«s  #4#.)  =  AB  vers  BA 


SIMPLE    CURVES    CONNECTING    RIGHT   LINES.          85 

For  BABl  =  2°  34'  this  becomes  AB,  -  AB  =  .WlAB  nearly. 
This  shows  that  ABi  exceeds  the  true  distance  measured  along 
the  base  by  only  one  thousandth  part  of  its  length  for  an  angle 
of  2°  34'.  If  greater  accuracy  than  this  is  desired,  the  angles 
between  the  auxiliary  line  and  the  base  line,  or  the  "courses," 
may  usually  be 'made  smaller  than  2°  34'.  Since  the  error  is 
approximately  as  the  square  of  the  number  of  minutes  in  the 
angle, 

for  1°  1?'     it  is  nearly  .00025,  or  nearly  1  in    4000; 

and  for         0°  38£'  "  "      "       .00006,   "       "       1  "  16000,  etc. 

To  find  the  angle  in  minutes  between  the  base  line  and  a  line 
joining  any  two  stations. 

Divide  the  difference  or  the  sum  of  their  departures,  according 
as  they  are  on  the  same  or  on  opposite  sides  of  the  base  line,  by 
their  distance  apart. 

Thus  the  line  BiDi  makes  with  the  base  line  the  angle 
PA  -  BB,   _  6240-3000  _          _ 
BD  WO 

The  line  AE,  makes  with  the  base  an  angle 


In  platting,  the  auxiliary  lines  are  penciled  only,  so  as  to  plat 
observed  objects  in  proximity  to  the  line  necessarily  observed 
from  the  auxiliary  lines.  When  these  objects  and  the  base  lines 
are  mapped  the  auxiliary  lines  need  not  be  retained. 

This  method  yields  quite  accurate  results  when  the  angles  be- 
tween tbe  lines  of  the  survey  are  2°  or  3°,  as  we  have  seen. 

For  perfect  accuracy,  however,  use  the  following  method  : 

Problem. — Having  run  a  broken  line  ABCD,  to  find  the  angle 
between  the  first  course  and  the  direct  course  JIT).  Represent  the 
lines  run,  in  their  order,  by  a,  &,  and  c,  and  the  deflection  angles  at 


FIG.  50. 

B  and  at  C  by  B  and  C  respectively.     Draw  CR  parallel  to  AB, 
and  (7/7  and  KDK  perpendicular  to  AB. 

Angle  DCR  =  DCE  -  RCE  ~  C  -  B, 


86 


FIELD-MANUAL    FOR   ENGINEERS. 


Now 


Now 

BH=bcosB;     HK  =  c  cos  (C  -  B) ;     AK  -  AB  +  BH+  HK 
CH  =  b  sin  B;     DR  =  c  sin  (C  -  B);    DK  =  CH  -  DR. 

DK 

4jr 

Also  AD  =  AK  -*-  cos  DAK. 

Of  course  the  method  is  applicable  whatever  the  number  of  lines 
run.      All  but  the  last  line  could  usually  be  taken  equal  to  a  whole 

number  of  chains,  which 
would  reduce  the  required 
computation  to  a  simple  mul- 
tiplication. 

E.  To  find  the  angle  of 
deflection,  i",  between  two 
straight  lines  A  V  and  VM, 
when  the  point  of  intersection 
is  inaccessible;  and  the  dis- 


A  M  tances  of  the  intersection  from 

FlG-  51-  given  points  on  the  lines. 

1.  Run  and  measure  a  perpendicular  PK  to  one  of  the  lines. 
Measure  the  angle  VPK  =  P.     Then 

V=  90°  +  P; 
KV=  A'Ptan  F, 
and  PV  -  KP  •  t- co*  P. 

2.  Run  and  measure  any  line  PL  from  one  line  to  the  other. 
Measure  also  the  angles  VPL  =  P'  and  PL  V  =i  /,.     Then 

V=  P'  +  L. 

Hence,  in  the  triangle  PVL,  PL  and  the  angles  are  known,  to 
find  FPand  VL. 

3.  If  obstructions  prevent  the  use  of  the  former  methods,  run 
and   measure  any  broken  line  ABCD.     Prolong  AB  and    BC  to 
meet  VM  at  M  and  N. 

Measure  the  deflection  angles  CBM  =  B,  DCN  =  C,  and  CDN 
=  D. 
Let  AMV  =  M,  and  CNV  =  N.     Then 


sTiHv  '  tto.(0+J5' 

=  BC  -f  CN.     Also     M  =  W  -  B-  C  -f  D  -  /?; 


SIMPI.I-;  CURVES  CUNXKCTI.NI.T  RIGHT  LINES. 


BM  = 


sin  N 


AM  =  AB  +  BM. 


Now  we  have  ^43/arid  the  angles  at  A  and  M,  to  find  vl  F,  MV, 
and  the  angle  F  =  A  4-  Jf.  A  similar  explanation  will  apply  to 
any  case. 

Wlie.n  a  broken  line  must  he  used,  the  above  method  involves 
fewer  computations  than  any  other. 

F.  To  locate  a  curve  joining  two  tangents  when  the  vertex  is 
inaccessible. 

Find   by  the  last    problem   the 
distances  Va  and  Vb  to  convenient 

points   on    the   tangents,    and    the  /  \    \,  V 

angle  V.  Then  assigning  or  com- 
puting the  tangent  T  =  A  For  BV 
from,  the  radius,  we  have 


and 


aA  =  T  -  a  V, 
bB  =  T-  bV. 


We  now  have  the  tangent  points 
and  can  run  in  the  curve  as  usual. 

GK  To  locate  a  curve  of  radius  U  or  tangent  T  when  the  vertex, 

the  beginning,  and  the  end  of  the 
curve  are  inaccessible.  Find,  as 
shown  with  Fig.  52,  the  angle  F 
and  the  distance  Fa  to  any  point, 
«,  on  A  V.  Then 

a  A  =  T  -  a  F, 
and 

cd        a  A 


FIG.  53. 


Also 

ac  =  Ad  =  J?versin  AOc. 
Drawing  the  tangent  cb,  we  have 

abc  =  AOc,     or     acb  =  90°  -  AOc. 

This  gives  the  direction  of  the  curve  at  c,  and  it  may  be  run  in 
each  way  from  c. 

To  pass  from  any  point  c  on  the  curve  to  any  point  n  on  the 
tangent. 


We  have 


1an  (dcu  =  an<<)  = 


88  FIELD-MANUAL   FOR   ENGINEERS. 

Set  the  instrument  at  c  and  turn  off  from  the  tangent  cb  an  angle 

ben  =  bed  —  den 
=  dbc  —  anc, 


,  Us  IV 

and  measure  en  =  —      — . 

cos  anc 

H.  To  find  any  desired  point  on  a  curve  when  obstacles  precl  udo 
the  use  of  ordinary  methods. 

(1.)  In  Fig.  28  measure  any  convenient  tangent  distance  AB  =  d. 
Then,  as  shown  in  Problem  1,  eq.  (43), 

cosec  A  =  — ;     then     t  —  d  tan  ±A. 

d  and  t  give  the  point  D  on  the  curve. 

It  is  important  to  note  that  if  AB  is  made  equal  to  one  half  the 
long  chord  for  any  number  of  stations  given  by  Table  IV,  BD  = 
AC  is  the  corresponding  middle  ordinate  and  may  be  found  in 
Table  V. 

Example. — Let  AD  be  a  4°  curve,  and  AB  —  one  half  the  chord 

of  four  stations  =  — ^—  =  199.35.     Then  Table  V  gives 

BD  =  13.94. 

(2.)  Problems  2  and  12  of  this  chapter  furnish  general  methods 
of  overcoming  obstacles  on  curves. 

(3.)  We  can  find  points  on  the  curve  as  follows: 
Let  b  be  a  station  near  the  obstacle.  Deflect  from  the  tangent 
at  b  some  small  multiple  of  the  deflection 
angle  for  one  station  ^D,  giving  the  line 
bd.  The  length  of  bd  may  be  taken  at 
once  from  Table  IV  and  measured  off, 
giving  d,  a  station  on  the  curve  beyond 
the  obstacle.  Taking  bm  =  \bd,  and 
measuring  off  the  middle  ordinate  me 
taken  from  Table  V,  gives  also  a  point  c 
on  the  curve. 

If  more  convenient,  make  bx  =  cm,  and 
xc  =  bm,  which  also  gives  c. 

Again,   run  the  tangent  b  V  =  d'  any 
jrIG  54  convenient  distance.     Then 


SIMPLE   CURVES   CONNECTING    RIGHT   LINES.          89 

Deflect  at  V  an  angle  equal  to  2bOV,  and  make  V'd  =  bV. 
d  will  be  a  point  on  the  curve.  The  number  of  stations  from  b 
is  equal  to 

bOd       2bOV 


bV  sliould  usually  be  taken  equal  to  a  whole  number  of  chains, 

7) 

in  which  case  —^-f  is  very  readily  found. 

The  lines  bV"  and  V"d  lying  on  the  inside  of  bd  may  be  run 
instead  of  bV  and  V'd. 


CHAPTER   V. 

LEVELING,  STADIA  MEASUREMENTS,  ETC. 

THE  field  operations  in  connection  with  the  level  are  more 
simple  than  those  required  with  the  transit,  but  they  require 
greater  skill  and  facility  in  manipulation  in.  order  to  produce 
correct  results. 

It  is  to  be  observed  that  the  elevation  of  points  is  a  relative 
matter.  The  elevation  of  some  point,  from  which  all  others  are 
to  be  found,  is  arbitrarily  assumed  to  be  100,  or  some  other 
number  sufficiently  large,  so  that  the  elevation  of  all  points 
to  be  considered  will  be  greater  than  zero. 

Near  the  coast,  and  in  fact  wherever  practicable,  it  is  im- 
portant to  refer  the  levels  to  the  mean  level  of  the  sea,  calling 
this  zero,  or  100,  or  some  other  number,  taking  care  to  estab- 
lish from  it  some  convenient  and  permanent  reference-point 
called  a  bench-mark,  or  bench. 

All  points  having  the  same  height  as  this  bench  are  some- 
times said  to  be  on  a  level  surface  called  the  datum.  This, 
however,  makes  no  difference  with  the  work  and  serves  no  use- 
ful purpose,  and  need  not  be  considered. 

All  elevations  thus  found  become  of  much  importance  in 
determining  the  relative  elevations  of  the  country,  and  in  the 
construction  of  physiographical  maps,  etc. 

Having  established  the  first  bench,  and  recorded  its  elevation, 
the  rod  man  stands  squarely  on  both  feet  behind  the  rod,  and 
rests  it  on  the  bench  as  nearly  in  a  vertical  position  as  possi- 
ble, which  is  best  done  by  simply  steadying  it  with  the  thumbs 
and  fingers,  taking  care  not  to  grasp  it. 

The  levelman  sets  up  his  level,  preferably  in  the  direction 
that  the  line  extends,  in  any  position  from  which  he  can  well 
see  the  bench,  as  well  as  points  to  be  afterwards  observed. 

He  then  makes  sure  that  the  instrument  is  in  adjustment, 
and  is  focused;  levels  it  carefully  and  sights  to  the  rod.  He 

90 


LEVELING,   STADIA    MEASUREMENTS,  ETC.  91 

may  plumb  the  rod  laterally  by  means  of  the  vertical  cross- 
wire  of  the  level,  and  the  rod  may  be  waved  gently  on  each 
side  of  the  vertical  toward  and  from  the  instrument,  the  short- 
est reading  being  the  true  reading. 

The  line  of  sight  on  the  rod  covered  by  the  horizontal  cross- 
wire  is  then  on  a  level  with,  or  at  the  same  height  as,  the  wire 
itself,  and  the  latter  is  therefore  higher  than  the  bench  by  the 
distance  intercepted  on  the  rod  between  the  line  of  sight  and 
the  bottom  of  the  rod.  This  is  called  the  reading  of  the  rod, 
or  simply  the  reading.  Adding  this  reading  to  the  height  of 
tho  bench,  we  obtain  the  height  of  the  cross-wire,  technically 
called  the  height  of  instrument,  and  designated  by  the  initials 
H.  I. 

Having  obtained  the  height  of  instrument,  the  elevation  of 
any  other  point  upon  which  the  rod  can  be  read  can  be  found 
by  taking  a  reading  of  the  rod  upon  it.  Of  course  the  point  is 
below  the  instrument  an  amount  equal  to  the  reading,  which 
must  therefore  be  subtracted  from  the  height  of  instrument  to 
give  the  elevation  of  the  point.  The  elevations  of  any  number 
of  points  may  be  thus  obtained. 

In  order  to  obtain  the  elevation  of  points  above  the  instru- 
ment, or  below  it  more  than  the  length  of  the  rod,  the  instru- 
ment must  be  moved  from  its  present  position  to  one  higher 
or  lower  as  the  case  may  require. 

Before  the  instrument  is  moved  to  a  new  position  a  temporary 
henfJt,  tailed  a  turning-point  (and  designated  by  T.  P.  or 
"Peg"),  must  be  established  and  its  elevation  ascertained  with 
care,  since  any  error  in  its  elevation  is  carried  forward  through- 
out the  whole  line  of  levels.  A  turning-point  must  be  firm 
and  definite  and  not  easily  disturbed  or  lost.  A  small  stake 
or  "  peg "  driven  with  its  upper  surface  about  flush  with  the 
surface  of  the  ground  is  generally  used.  The  top  of  a  rock  may 
well  serve  the  purpose. 

Benches  and  turning-points  are  of  course  the  same  in  prin- 
ciple, but  the  more  or  less  permanent  point  taken  as  the  basis 
of  the  elevations  of  the  Survey,  and  also  those  made  usually 
along  and  near  the  line,  for  future  reference,  whether  used 
as  turning-points  or  not,  are  usually  called  benches. 

From  this  new  turning-point  we  proceed  precisely  as  before, 
by  getting  a  new  height  of  instrument,  etc.,  and  it  is  important 


FIELD-MANUAL    FOR    ENGINEERS. 


to  note  that  the  operation  just  described,  of  obtaining  a  height 
of  instrument  from  a  bench  or  turning-point,  and  then  obtain- 
ing the  heights  of  any  number  of  desired  points  within  range 
of  the  instrument,  including  a  new  bench  or  turning-point, 
includes  the  whole  subject  of  leveling. 

Since  the  cross-wires  must  be  higher  than  any  point  upon 
which  a  reading  is  taken  it  must  be  remembered  that: 

1.  The  reading  on  a  point,  added  to  its  elevation,  gives  the 
height  of  instrument. 

2.  The  reading  on  a  point  subtracted  from  the  height  of  in- 
strument gives  the  elevation  of  the  point. 

In  other  words:  We  must  add  a  reading  (to  the  height  of 
some  point)  to  get  a  height  of  instrument,  and  must  subtract 
a  reading  (from  a  height  of  instrument)  to  get  the  height  or 
elevation  of  some  point. 

The  theory  of  leveling  requires,  therefore,  only  a  simple 
application  of  addition  and  subtraction,  and  it  is  not  easy,  it 
would  seem,  to  go  wrong  in  it. 


Station. 

+  8 

H.  I. 

-  S 

Elevs. 

Remarks. 

EM 
0 
1 
Peg        2 

3.46 

203.46 

7.29 
5.34 

0.81 

200.00 
196.17 
198.12 
202.65 

W.  Oak  60  ft.  R.  of  Station  0 

+  40 
8 
4 
5 

Peg  +  60 
6 

4.17- 
6.18 

206.82 
211.93 

1.12 
3.16 
6.09 
4.14 
1.07 
3.13 

205.70 
203.66 
200.73 
202.68 
205.75 
208.80 

The  accompanying  table  shows  a  convenient  form  of  field- 
book  for  keeping  the  level  notes  of  a  railway  or  other  survey. 
The  first  column  contains  the  stations  and  benches.  The  second 
the  plus  readings  taken  on  points  whose  elevations  are  assumed 
or  already  determined.  The  third  column  contains  the  heights 
of  instrument  recorded  one  line  below  the  elevation  of  the  turn- 
ing-point (or  bench)  from  which  it  is  calculated.  The  fourth 
column  contains  the  minus  readings.  The  fifth  column  con- 
tains the  elevations  of  all  points  observed.  The  right-hand 
page  is  reserved  for  remarks  describing  the  benches  and  their 
location,  also  objects  crossed  by  (or  near)  the  line,  as  roads, 
streams,  ditches,  etc. 


LEVELING,  STADIA    MEASUREMENTS,  ETC.  93 

It  is  to  be  observed  that  for  any  series  of  levels  the  sum 
of  the  plus  sights  less  the  sum  of  the  minus  sights  (omitting 
those  for  determining  intermediate  points  on  the  ground)  is 
equal  to  the  difference  between  the  first  and  last  elevation. 

Thus  to  prove  station  3,  we  have 

3.46  -f  4.17  —  0.81  —  3.16  =  203.66  —  200  —  3.66. 
To  prove  the  H.  I.,  211.93,  we  find 
3.46  -f-  4.17  4-  6.18  —  0.81  —  1.07  =  211.93  _  200  —  11.93. 

In  practice  it  is  best  to  check  each  page  of  the  field-book  by 
comparing,  as  above,  the  first  turning-point  or  height  of  instru- 
ment (brought  over  from  the  preceding  page),  with  the  last 
turning-point  or  height  of  instrument  on  the  page. 

To  facilitate  this  work  some  engineers  use  two  columns  for 
the  minus  sights,  placing  those  which  determine  the  turning- 
points  in  a  column  by  themselves. 

This  practice  is  commendable. 

Benches  should  be  established  at  short  distances  apart  along 
the  line,  taking  care  to  locate  them,  so  far  as  possible,  near  the 
crossings  of  roads,  streams,  railways,  etc.,  and  at  all  points 
where  their  need  can  be  foreseen,  in  the  location  of  cattle- 
guards,  culverts,  bridges,  etc.  Of  course  an  extra-good  bench 
should  be  established  at  the  end  of  the  survey.  An  extra-good 
turning-point  or  bench  should  also  be  established  at  the  end 
of  each  day's  work. 

The  object  of  obtaining  a  line  of  levels  is-  to  furnish  a  profile 
of  the  line  surveyed,  showing  the  undulations  of  the  surface 
over  which  it  passes. 

The  elevations  are  platted  on  profile  paper,  the  horizontal 
scale  being  about  400  feet  to  an  inch,  and  the  vertical  scale 
about  25  feet  to  an  inch.  This  distortion  of  scale  magnifies  the 
vertical  measures  about  ****/.,?,  =  16  times,  so  that  the  slight 
changes  in  the  elevation  of  the  surface  may  be  distinctly  seen. 

In  running  a  line  of  "  flying  "  levels  no  readings  are  taken 
except  on  turning-points.  If  the  difference  of  levels  of  the  ex- 
treme points  only  is  desired,  it  is  necessary  to  find  the  differ- 
ence only  between  the  sum  of  the  plus  and  of  the  minus  read- 
ings, as  already  explained.  This  is  very  convenient  for  testing 


94  FIELD-MANUAL  FOR  ENGINEERS. 

a  line  of  levels  already  run;  in  which  ease  it  is  best  to  touch 
on  the  benches  only,  and  if  found  correct,  the  intermediate  ele- 
vations may  be  regarded  as  correct  also. 

No  line  of  levels  should  be  taken  as  correct,  and  so  used, 
without  first  being  carefully  checked. 

The  Philadelphia  rod  is  the  most  convenient  and  best  rod 
in  use.  It  is  plainly  lettered  and  easy  to  use,  and  may  be 
read  by  the  levelman  when  desirable,  and  at  a  distance  of  sev- 
eral hundred  feet. 

To  Locate  a  Level  Line.— Set  a  peg  at  the  desired  height,  as 
a  starting-point,  and  take  a  reading  of  the  rod  thereon.  Send 
the  rod  forward  in  the  desired  direction,  and  have  it  moved  up- 
ward or  downward  along  the  slope  of  the  ground  until  a  point 
is  found  which  gives  the  same  reading  as  before. 

Of  course  the  reading  is  taken  on  a  peg.  This  second  peg 
is  of  the  same  height  as  the  first.  Find  in  the  same  way  :i 
third  peg  from  the  second,  etc. 

In  this  way  stakes  may  be  set  at  points  on  the  ground  level 
with  the  top  of  a  proposed  dam,  or  with  the  supposed  top 
of  water  flowing  over  the  dam.  Then  joining  these  stakes  by 
lines,  the  area  thus  inclosed  may  be  measured. 

The  water  behind  a  dam  is  not  level,  but  is  curved  con- 
cavely  upward  and  so  increases  in  height  back  of  the  dam, 
and  sets  back  farther  than  if  level. 

For  the  subject  of  backwater,  Works  on  Hydraulics  must  be 
consulted. 

Other  applications  of  the  level  line  are  to  obtain  "  contour 
lines "  for  topographical  maps,  for  levees  in  irrigated  rice- 
fields,  etc. 

To  Run  a  Grade-line This  consists  in  setting  a  series  of 

pegs  so  that  their  tops  shall  be  points  in  a  line,  which  shall 
have  any  required  slope  ascending  or  descending. 

First  drive  pegs  at  each  end  of  a  line  to  the  heights  required. 
These  heights  may  differ  by  a  given  amount,  or  this  difference 
may  be  undetermined. 

Set  the  level  over  one  of  the  pegs  and  measure  the  height,  a.  of 
the  cross-wires  above  the  top  of  the  peg. 

Set  the  rod  on  the  other  peg,  and  make  the  reading  on  the  rod 
equal  to  the  height  a. 

Without  disturbing  the  level  drive  any  desired  number  of  pegs 


LEVELING,  STADIA   MEASUREMENTS,  KTC.  95 

along  the  line,  so  that  the  reading  on  each  will  also  be  equal 
to  a. 

A  line  of  uniform  grade  or  slope  is  not  a  straight  line. 

Calling  the  globe  spherical,  this  line  when  traced  in  the  plane 
of  a  great  circle  would  be  a  logarithmic  spiral.  On  a  length  of 
six  miles  the  distance  of  its  middle  point  from  the  middle  of  its 
straight  chord  would  be  six  feet  almost  exactly. 


CORRECTION  FOR  THE  EARTH'S  CURVATURE  AND  FOR 
REFRACTION. 

This  is  necessary  for  long  distances. 

Let  AB  (Fig.  55)  represent  a  portion  of  a  section  of  the  earth's 
surface.  Then  if  a  level  be  set  at  A,  the  line  of  sight  of  the  level 
will  be  the  tangent  AD,  while  the 
true  level  will  be  the  arc  AB.  The 
difference  BD  between  the  line  of 
sight  and  the  true  level  is  the  cor- 
rection for  the  earth's  curvature  for 
the  distance  AB.  This  must  be  sub- 
tracted from  the  reading  of  the  rod 
at  B,  or,  what  is  the  same  thing, 
added  to  the  height  of  B,  as  given 
by  the  reading  of  the  rod. 

Let  AE  =  R,  AB  =  1),  and  BD 
=  E.     By  geometry, 


AD*  =  BD(BD  4- 
A& 


BD  = 


FIG  55. 


BD  + 


Omitting  BD  in  the  right-hand  member,  since  it  is  small  com- 
pared with  2R,  and  supposing  AD  —  AB  =  J),  we  obtain 


22t       2  X  20913650 


rx  =  .0000000239087)5.    . 


(1) 


This  formula  gives  a  result  or  value  for  E  slightly  too  small; 
but  the  relative  error  is  only  about  one  in  24,000  for  a  distance  of 


^Q  =  39.6  miles,  or  arc  of5-    m 


=  34'  22".65. 


96  FIELD-MANUAL    FOR    ENGINEERS. 

In  observing  distant  objects,  a  ray  of  light  traversing  tlie  air 
from  an  object  to  the  eye  or  instrument  is  refracted,  and  takes  a 
curved  path  which,  for  points  near  the  surface  of  the  earth,  is 
practically  the  arc  of  a  circle,  concave  downward,  and  whose 
radius  is  7.K. 

Thus  a  point  at  C  (Fig.  55)  would  appear  at  D  higher  than  it 
really  is  by  an  amount  CD.  This  may  be  found  from  the  above 
formula  by  substituting  7R  for  2L 

Hence  the  correction  for  refraction  is 


E'  =  ~_  =  .  00000000341  5D2  .....     (2) 

The  correction  for  curvature  and  refraction  is 

IP        D2        3  /)2 
E"  =  £C=BD-CD=:  —  -  ~—rt  =  -  -    =  .000000020492D8.  (3) 


This  must  be  added  to  the  apparent  elevation  of  the  observed 
object  to  give  the  true  elevation. 

Table  XI  gives  the  value  of  the  correction  for  the  value  of 
R  =  20911790  feet. 

When  it  is  possible  to  set  the  level  midway  between  the  points 
whose  heights  are  required,  the  corrections  will  balance  each 
other  and  may  be  omitted. 

The  above  equations  may  be  put  into  a  form  sometimes  more 
convenient  as  follows  : 

The  length  of  arc  on  the  earth's  surface  subtending  angle  of 
one  minute  is 


Then  ~  =  .12638 (!') 

=  the  correction  for  refraction  for  distance  6083  ft.  or  arc  of  1'. 
Also  ~~    =  .8846 (2') 


=  the  correction  for  curvature  for  the  same  distance  or  arc  1', 


LEVELING,  STADIA   MEASUREMENTS,  ETC, 


97 


and 


i   1  .75828 (3') 

7/t 


=  the  correction  for  curvative  and  refraction. 


TRIGONOMETRIC  LEVELING. 

First  Method. — When  the  point  C  (Fig.  56)  can  be  seen  from 
two  points  A  and  B  on  the  same  level,  then 

AD  -  CD  cot  CAD,     and    BD  =  CD  cot  CBD, 

Subtracting  gives 

AB 

AB=OD(cot  CAD-  cot  CBD),  or  ^=calCAD_^cB1)-      W 

Second  Method. — Let  A  and  B  (Fig.  57)  occupy  any  positions 

C 


B 
FIG.  56. 


except  in  line  with  C.     Measure  AB  and  the  angles  at  A  and  B\ 
also  the  angle  of  elevation  GAD. 


C=1SQ°-A-B,  AC  = 


sin  C 


,  and  CD  =  AC  sin  CAD.  (5) 


If  A,  B,  and  C  are  in  the  same  vertical  plane,  the  solution  is  in 
no  wise  affected. 

Unless  the  distance  AD  =  D  is  short  it  is  necessary  to  add  to. 
the  correction  found  by  the  preceding  formulas  the  correction 

O   7^0 

for  curvature  and  refraction,  namely,  —  —  —  .0000000204927)2, 


FIELD-MANUAL    FOR    ENGINEERS. 

To  find  the  height  of  instrument  by  an  observation  of  the  horizon 

(Fig.  58). 

First  Method. — Let  C  be  the  place  of  the  transit,  and  BAA  a 

portion  of  a  section  of  the 
earth's  surface. 

Were  there  no  refraction 
the  line  of  sight  would  be 
the  tangent  CA\  ACE—  C 
would  be  the  angle  of  de- 
pression or  dip  ;  and  we 
would  have 

BC  =  BOX  exsec  CO  A 
=  It  exsec  C. 

Owing  to  refraction,  how- 
ever, the  line  of  sight  would 
be  a  curve  concave  down- 
ward whose  radius  =  7/£; 
and  it  would  therefore  ex- 
tend from  C  to  a  point  A', 
say  a  distance  A  A'  beyond 
A. 

Draw  CDF  tangent  to  this 
curve  at  C  to  meet  the  ra- 
dius AO  prolonged  in  F. 
Draw  also  the  tangent  A't. 

Let  COA=N,  COU=I1', 
and  CO  A'  =  0.  Then 


FIG.  58. 


CO  =  r  sec  //  = 


cos  H' 


Let  E,  on  A'O  prolonged   but  not  shown,  be  the  center  of  the 
arc  A'C.     Now 

#0V_  CO9 


cos  0  =  -  cos  COE  =  - 


2CO .  EO 


-  ___ 

TsT^eclT^  12 

Clearing,    substituting   1  —  vers  0  for   cos  0,    1  —  vers  H  for 
cos  II,  and  1  -f  exsec  7?"  for  sec  H,  we  find 


13  vers  0  —  13  vers  //  +  exsec  Jf. 


LEVELING,  STADIA    MEASUREMENTS,  ETC.  99 

Or,  writing  versines  for  exsecants  or  vice  versa,  we  have 

vers  0  =  £  vers  II, 
or,  approximately, 

exsec  0  =  £  exsec  H. 

In  the  triangle  CEO, 

49  r5  -f  36  ?•*  -  ~U6l  _  85r2  -  r2  sec9  H  _  85  -  sec2  H 
2~x~  6r"x~7V  *  84?"  ~~84~~ 

Also  sin  E  =  sin 


.-.  sin  #sin  0  =  sin2  0'  ----  =  (1  —  cos'2  Oj—  —  — 

_  1  70  -  169  cos2  H  -  sec2  H        sec  H 

~l44~  ~~T"~ 

Now  since  OA'  is  tlie  prolongation  of  EO,  and  JS'C'and  OZ)  are 
perpendicular  to  CD  and  therefore  parallel,  we  have 

CO  A    -  1)0  A'  =  COD,     or     0  -  E  =  II'; 
.-.  cos  H'  =  cos  (0  —  E)  =  cos  0  cos  E  -\-  sin  0  sin  E. 

Substituting  in  this  the  above  general  values  of  cos  0,  cos  E, 
and  sin  7?  sin  0,  and  expanding  and  reducing,  we  find 


13  cos  E  +  sec  H  ,    (sec  H  —  cos  //) 


Putting  cos  H=  1  —  vers  //,  sec  7/=  1  +  exsec  IT,  etc.,    we 
find 

(exsec  //  —  vers  H) 
vers  /f  =  f  vers  77  --         —  —  —        —  ,     .     .     (a) 

or  vers  77'  =  |  vers  77,  very  nearly,  ......     (6) 

or  exsec  //'  =  f  exsec  //,  very  nearly  ......     (7) 

exsec  H  '  —  vers  //' 

Hence       vers  //'  =  f  vers  H  --  —  . 

i& 

From  this  we  have 

,   £(exsec  H'  —  vers  H'} 
vers  H  —  \  vers  //'  -f  ^  —  ,     .      (ft) 


100  FIELD-MANUAL   FOR    ENGINEERS. 

or  vers  //=  |  vers  //',  very  nearly,      .....     (6') 

or  exsec  H=  |  exsec  II',  very  nearly  ......     (7') 

Since  the  versines  of  small  angles  are  very  nearly  in  the  ratio 
of  the  squares  of  the  magnitudes  of  the  angles,  we  have,  from  (6), 


H—  4/JZT  =  1.08//',     approximately.    .     .     .     (?') 

T 

Supposing  r  =  4000  miles  and  AB  =  —  —  50  miles,  then    it  is 

80 

easy  to  show  that  the  error  of  eq.  (7)  is  less  than  .00000002,  and 
the  error  of  (6)  is  about  .0000000004. 

Ki'<n/'iple  1.  —  The  observed  dip  of  the  sea  horizon  is  //'  .—  24'. 
What  is  the  height  of  the  instrument  above  the  sea? 

We  have 

BC  =  r  exsec  H-  r|  exsec  //'  =  20914000  X  .00028467 

'=  595.35  feet,  exactly. 

Example  2.—  Let  r  —  4000  miles,  and  AB  =  50  miles.  What  is 
the  observed  angle  of  depression  H',  and  what  is  the  height  of 
the  observer  above  the  sea  ? 

We  have 

Kf\ 

H=  j^  X  57°.29578  =  0°.7162  =  42'  58". 

.-.  H'  =  0.716  -4-  1.08  =  0°.663  =  39'  47". 

Also, 

7i  =  r  exsec  H=  4000  X  .0000781  =  .3124  miles  =  1649.47  feet. 

The  exact  relations  between  II  and  H',  shown  above,  would 
seem  to  be  more  satisfactory  than  the  approximate  equations  in 
general  use  even  if  these  were  regarded  as  sufficiently  accurate. 

THE  STADIA. 

The  stadia  is  a  compound  cross-wire  ring  or  diaphragm  having 
three  horizontal  wires. 

The  two  outer  ones  are  called  stadia  wires,  and  distances  deter- 
mined by  means  of  them  are  called  stadia  measurements. 


LEVELING,  STADIA    MEASUREMENTS,  ETC.  101 

The  stadia  wires  are  adjusted  so  as  to  intercept  a  certain  space 
on  a  rod  at  a  given  distance  from  the  transit  and  perpendicular  to 
the  line  of  sight. 

Let  C  (Fig.  59)  =  the  distance  of  the  object-glass  from  the  axis 
of  the  transit,  and/  =  the  focal  length  of  the  object-glass. 


FIG.  59. 

This  focal  length  is  equal  to  the  distance  of  the  cross-wires 
from  the  object-glass  when  this  is  focused  for  a  distant  object. 

This  focal  length  may  be  found  also  by  removing  the  ohject- 
glass,  exposing  it  to  the  rays  of  the  sun,  and  noting  at  what  dis- 
tance from  the  center  of  the  glass  the  rays  form  a  perfect  and 
minute  image  of  the  sun  on  a  smooth  surface. 

Let     Cm  =  I',     Cn  =  I,     DE  =  S',     and    HK  =  8. 

The  focal  distance  OF  is  constant,  but  Co  =  c  varies  with  the 
position  of  the  object-glass,  and  hence  CF  is  also  variable. 

Let  Co  =  c'  when  the  rod  is  at  DE,  and  Co  =  c  when  the  rod  is 
at  HK. 

Now,  from  the  figure, 

* 

' 


—  - 

Fm  ~  DE'  I'  -  (c'  +7) 


8'  is  usually  assumed  =  1  foot,  and  Fm  =  I'  —  (c'  -\-  f)  —  100 
feet;  and  the  stadia  wires  are  then  adjusted  accordingly. 

c'  is  measured  on  the  telescope  when  the  object-glass  is  focused 
on  the  rod  at  the  assumed  distance. 

To  measure  any  other  distance  the  rod  is  again  observed  at  the 
desired  point  and  the  space  8  noted,  which  placed  in  (8)  gives 
I  —  (f'  +/)  —  ^i»  sav-  We  may  then  measure  c  on  the  telescope. 
Then 


102 


FIELD-MANUAL   FOE   ENGINEERS. 


Since,  however,  c  has  but  a  small  range  of  values,  it  will  usually 
be  sufficient  to  assume  it  to  be  constant  and  equal  to  some  mean 
value. 

Suppose  that  in  (8)  c  =  c'  =  Ci,  and  solving  we  find 

SI'  -  S'l 


If  we  observe  S'  and  8  corresponding  to  any  two  distances  I' 
and  I  and  substitute  in  (9),  we  have  Ci  -(-/. 

Having  found  c,  +/,  lay  off  Cm  =  100  +  c,  -f  /,  or  Fm  =  100, 
and  adjust  the  stadia  wires  to  subtend  DE  =  just  one  foot  at  that 
distance. 

Then  from  (8),  writing  c,  for  c  and  c',  we  have 


or,  omitting  accent, 


(10) 


Example.  —  Suppose  at  £'  =  100  we  find  S'  =  1,  and  at  I  =  500 
we  find  S  =  5.0453. 
Then  eq.  (9)  gives 

504.53-500  _ 
~ 


Then,  from  eq.  (10),  I  =  100$  -f-  1.12,  provided  the  stadia  wires 
are  spaced  so  as  to  intercept  1  foot  at  101.12  feet  distance  from 
the  center  of  the  instrument. 

The  foregoing  formulas  must  be  modified  when   the  line   of 

collimation  is  oblique  to  the 
horizon,  which  is  usually  the 
case.  Thus,  in  Fig.  60,  let 
jyE'  =  the  space  intercepted 
on  the  rod  when  the  line  of 
collimation  FH  is  horizontal, 
and  let  DE  =  S  =  the  space 
intercepted  when  the  line  of 
collimation  Fn  makes  an  angle  nFH  =  a  with  the  horizon. 
Let  DFE  =  D'FE'  =  0. 


Ff0 


LEVELING,  STADIA   M-EASUREMENTS,  ETC.  103 

In  Fig.  60, 

8  =  DH  -  EH  =  HF  [tan  (a  +  |0)  -  tan  (a  -  £0)]. 
The  horizontal  reading  desired  is  D'E'  =  2JBFtan  £0. 

D'^'  2  tan  40 

Dividing  gives      — ^—  =  —  — r^. 

8          tan  (a  -f-  £0)  —  tan  (a  —  £0) 

2  sin  40 
But  2  tan  40  =  - 

COS  ^U 

Also,  by  Chapter  III,  formulas  (26),  (27), 

sinO 


tan  (a  -f  £0)  —  tan  (a  —  £6)  = 


cos  (a  -j-  -|0)  cos  (a  —  i0) 

2  sin  |0  cos  4.0 
cos8  a  —  sin2    0* 


Substituting  these  values,  we  obtain 

D'E'        cos-  a  —  sin2 


cos2 


(11) 


If  we  neglect  sin2  |0,  or,  what  is  equivalent,  add  sin8  ^0  to  the 
numerator,  we  introduce  a  relative  error  of 

sin2  40  sin2  4,0 

— —  —  —  =  sin*  |0  sec2  a,  very  nearly. 

cos8  a  —  sm-  ^^         cos^  a 

Again,  if  we  add  sin*  |0  to  the  denominator,  making  it  unity, 

we  introduce  a  relative  error  of  — — ^-r  =  sin2  40  sec2  40. 

cos2  |0 

The  first  of  these  errors  increases  the  fraction,  and  the  second 

decreases  it.     Moreover,  since  0  =  — —  —  —  34'   22". 65,    we 

J(JO 

have  in  all  practical  cases  a  >  |0,     or     sec2  a  >  sec2 10. 

Hence  the  fraction  is,  by  the  double  approximation,  increased 
sliglitly  more  than  it  is  decreased.     Hence 

TV  W  T)'  W 

— —  <  cos8  a  ;  but  —    —  —  cos*  a,  almost  exactly.   .     (12) 

o  o 

The  total  relative  error  is 

e  —  sin2  ^0  (sec2  a  —  sec2  4;0) 
=  sin3  £0  (tan2  a  —  tan2  £0). 


104  FIELD-MANUAL   FOR   ENGINEERS. 

Since  tan1  ^Q  is  very  small  compared  with  tan2  a,  we  have 
e  =  sin5  -|0  tan-  a,  very  nearly, 

=  .000012  tan2  a,  very  nearly. 
Hence 

D'E' 


=  cos5  a(l  -  .000012  tan2  a) 

o 

=:  cos2  a  —  .000012  sin8  a (13) 

Since  sin  £9  =  tan  ^0  =  — —  =  .005.  very  nearly,  the  quantity 

neglected  above  is  only  (.005)4  =  .00000000625  and  does  not  come 
within  the  range  of  the  table. 

The  above  is  the  coefficient  of  reduction  by  which  to  multiply 
the  observed  space  DE  —  S  in  order  to  produce  the  true  space 
D'E'  which  would  be  observed  at  the  same  distance  if  the  line  of 
colliination  were  horizontal. 

Hence  we  have,  from  (10), 

I  =  (1005  +  c  +/)(cos2  a  -  .000012  sin2  a). 

By  eqs.  (25)  and  (26),  Chap.  Ill, 

1  4-  cos  2a 
cos2  a  =  ~  —  SB  .  J  • 


and  Bin'  a  = 

Hence 
I  =  (100S+  «  +/)l  -  ~-  -  .000013  ,  .     (14) 


I  =  (100/8  +  c  +/)  (l  -   ver*2(l  \  very  nearly.     .     .     .    (14') 

\  / 

These  coefficients  may  be  read  from  a  table  of  versed  sines 
without  any  computation  whatever. 

The  last  equation  is  quite  accurate  enough,  but  the  coefficients 
of  Table  XIII  are  calculated  by  the  exact  formula. 

JSxample. — Find  the  coefficient  for  a  =  7°  20'. 

We  write  half  the  vers  14°  40' =  .01629 

and  subtract  from  unity  and  find  coefficient —  .98371 

Another  method  of  procedure  is  that  in  which  the  rod  is  held 
perpendicular  to  the  line  of  collimation. 


LEVELING,  STADIA   MEASUREMENTS,  ETC. 


105 


To  secure  this  position  of  the  rod  a  bar  is  attached  to  it  having 
sights  upon  it,  through  which  the  rodinan  watches  the  instrument 
during  an  observation,  the  line  of  sights  being  perpendicular  to 
the  rod. 


FIG.  61. 
The  horizontal  distance  of  the  point  B  from  the  instiument  is 

IE  =  IK  +  KH  =  1m  cos  a  +  Bm  sin  a, 
or  IE  =  (1005  +  c  +/)  cos  a  +  r  sin  a. 

(See  Fig.  61,  in  which  r  is  the  reading  of  the  rod  by  the  line  of 
collimation.) 

The  elevation  of  B  above  /is 

BH  =  mK  —  Bm  cos  a, 
or  BH  =  (1008  +  c  -f  /)  sin  a  -  r  cos  a.    .     .     .     (15) 

When  the  distances  are  sufficiently  great  correction,  must  be 
made  for  curvature  and  refraction  as  already  pointed  out. 

THE  GRADIENTER. 

This  attachment  consists  of  a  screw  working  against  a  clamping- 
arm  suspended  from  the  horizontal  axis,  on  the  opposite  end  from 
the  vertical  arc. 

A  strong  spiral  spring  presses  the  arm  against  the  end  of  the 
screw.  The  large  silvered  head  of  the  screw  is  usually  divided 
into  100  equal  spaces.  When  the  screw  is  turned  the  head  moves 
along  one  division  of  a  small  silvered  scale  for  each  revolution  of 
the  screw. 

When  the  regular  clamp  of  the  telescope  is  free  the  telescope 
may  be  revolved;  but  when  the  telescope  is  held  by  this  clamp  it 


- 


=  -  r     -    = 


2     _  -  -    '    -  I  - 


.  -    - 


.   ' 


<K> 


Let 

-i     ^ 


,-----      -T 


€» 


^  , 

-* 


^    ::  -  >.  . 


.,    j         -  — 


: 
-     - 


108 


FIELD-MAKUAL   FOR   ENGINEERS. 


F,  and  the  horizontal  lines  AK  and  CO  to  meet  BF  in  .STand  0. 
Join  AB,  and  drop  the  perpendiculars  aimibi  and  a2?wa&2.  Let 
7i  =  the  number  of  stations  in  AC  or  CB.  Then,  since  these  dis- 


tances are  measured  horizontally,  we  have  AH  =  HK,  and  there 
fore  AD  =  DB. 

The  vertical  line  CD  is  therefore  a  diameter  of  the  parabola, 
and  the  distances  of  points  on  the  curve  in  a  vertical  direction 
from  corresponding  points  on  the  tangent  AF  are  in  the  ratio  of 
the  squares  of  the  distances  of  these  points  from  A.  Also, 


.:     CE=ED. 
Now,  since  CF  =  CA, 

FO  =  CH=  the  rise  of  AC  in  n  stations  =  ng. 
Also,  OB  =  the  fall  of  OB  in  n  stations  =  ng'  ; 

.'.     FB  =  n(g+g'}. 
Now,  since  B  is  %n  stations  from  A,  we  have 

-r-r  =      ;  9  , 
4n*  4n 


Offset  at  first  station  from  A  —  a,  mi  =  a  = 


(18.1 


The  value  of  a\m,i  being  determined,  the  distances  of  the  curv<< 
at  all  points  from  the  tangent  ^L^are  also  known.     Thus 

a^m-2  =  4a,     Ce  =  9a,  etc. 

It  is  easy  to  calculate  the  heights  of  the  points  of  the  curve 
above  AK,  though  these  heights  are  scarcely  needed.     Thus 

=  g  —  a  ; 


EH  =  CH  -  CE 


-  9a,  etc.,  etc. 


LEVELING,  STADIA  MEASUREMENTS,  ETC.        109 

Finally, 

BK  =  FK-  FB  =  2ng  —  ±n*a  =  2ng  -  n(g  +  g'}  =  n(g  -  g'}, 
or 

BK=  CH-  OB  =  ng  -  ng'  =  n(g  —  g'}  as  before. 

The  successive  grades  are  found  by  taking  the  successive  differ- 
ences of  the  heights  just  found.  Thus 

mibi—o  =  g—  a  ;  m^—mibi  =  g—  3«  ; 

The  change  in  the  grades  from  station  to  station,  which  is  the 
same  as  the  chord  deflection  of  the  curve,  is,  we  observe,  equal  to 
the  constant  quantity  2a. 

Second  Proof.  —  The  curve  must  change  its  direction  in  its 
length  of  2n  stations  an  amount  equal  to  g  ±  g',  and  therefore 

in  each  station  it  must  change  an  amount  equal  to  —  —  —  ;   and 

2n 

this  is  equal  to  2a  by  eq.  (18). 

Third  Proof.—  The  ordinates  to  the  curve  are  a,  4«,  Qa,  16«,  etc. 

The  differences  of  ordinates  are  3a,  5a,  7«,  etc. 

.-.  the  differences  of  grades  are  2a,  2a,  etc.,  a  constant. 

Fourth  Proof.  —  Let  m2  represent  the  nth  station,  and  prolong 
the  chord  m^m-i  to  meet  CE  in  x  (not  shown). 


Now  awi  =  (n  —  1)X     a^m*  =  tfa,     and     CE  —  (n 
But 
Cx  +  a1ml  =  2a-tm-l,     or     Cx  =  2a^my  —  a-^m^  =  [(n  -j-  I)9  —  2]a. 

Hence  Ex  =  CE  —  Cx  ~  2a. 

In  finding  the  value  of  a,  etc.,  it  is  necessary  to  know  when  we 
are  to  take  the  sum  of  the  grades,  and  when  the  difference;  for 
there  may  be  four  combinations  of  two  adjacent  grades. 

It  is  only  necessary  to  observe  : 

(1)  That  when  the  grades  are  both  ascending  or  descending  FB 
is  equal  to  n(g  —  g'},  g  being  the  steeper  of  the  two  grades,  and 
therefore 

_  n(g  -  g')  _  g_--_jf_ 
W  in    ' 


110  FIELD-MANUAL   FOR  ENGINEERS. 

(2)  That  when  one  of  the  grades  is  ascending  and  the  other  de- 
scending, 

FB  =  n(g  +  g'),   and     a  = 


4n 

If  one  of  the  tangents  is  horizontal,  let  g  =  grade  of  the  other 

and  then  a  =  ~-. 
4n 

In  all  cases  it  is  necessary  to  consider  only  the  change  of  grade 
at  C;  which  we  will  represent  by  G.     Then 

&  =  ff  -  ff',     or     G-  =  g  +  /, 

according  as  the  grades  are  both  rising  or  falling,  or  one  rising 
and  the  other  falling.     Then 

FB        G 

rB  =  JiG,   and     a  =  —  -  =  — 
4n?        4n 


These  formulas  apply  without  change  to  points  between  stations, 
as  the  following  examples  will  illustrate. 

Example  1. — A  .9$  up  grade  joins  a  .3$  down  grade  at  station 
76  at  an  elevation  of  94.0.  Find  ordinates,  etc.,  for  a  curve  6  sta- 
tions long. 

a  =  '9^/3  =  .1.     (See  Fig.  64.) 

Stations  on  AF 73  74  75  76  77  78  79 

Elevations  on  AF 91.3  92.2  93.1  94.0  94.9  95.8  96.7 

Corrections 00        .1        .4        .9  1.6  2.5  3.6 

Elevations  on  curve....  91.3  92.1  92.7  93.1  93.3  93.3  93.1 

It  is  necessary  to  compute  the  ordinates  a,  4a,  etc.,  for  one  half 
the  curve  only,  since  they  are  the  same  for  corresponding  points 
on  each  tangent.  Thus  we  have  : 

Stations  on  AC  and  CB  73  74  75  76  77  78  79 

Elevations  on  AC  and  CB  91.3  92.2  93.1  94.0  93.7  93.4  93.1 

Corrections 0.0  .1  .4  0.9         .4         .1       0.0 

Elevations  on  curve 91.3  92.1  92.7  93.1  93.3  93.3  93.1 


LEVELING,  STADIA   MEASUREMENTS,  ETC, 


111 


Example  2. — A  .2$  up  grade  is  continued  beyond  station  17, 
whose  elevation  is  46.4,  by  an  .8$  up  grade.  Find  ordinates,  etc., 
for  a  curve  6  stations  long. 


Stations  on  A  C  and  CS. .  14  15        16        17        18  19        20 

Elevations  on  AC  and  CB  45.8  46.0     46.2     46.4     47.?  48.0     48.8 

Ordinates 0.0  .05       .20       .45       .20  .05    0.0 

Elevations  on  curve 45.8  46.05  46.4    46.85  47.4  48.05  48.8 

To  find  the  grade  of  the  parabola  at  any  point. 
Let  the  curve  be  that  of  Example  1,  and  let  the  point  be  at  a\ 
-f  40,  or  140  feet  from  A. 
Elevation  at  a,  -f  40  =  92.2  -f  .40X.9  =  92.2+  .36  =  92.56. 


Also, 


(1.4}2a  =  I  Ma  =  .196  =  .20,  nearly. 


Therefore  elevation  of  curve  at  a,  -f  40  =  92. 86. 

The  special  need  of  these  curves  is  in  sags  in  the  grades  to  pre- 
vent the  breaking  of  the  train. 

According  to  Wellington,  Railway  Location,  page  365,  vertical 
curves  in  sags  should  be  at  least  200  feet  long,  or  100  feet  on  each 
side  of  the  vertex,  for  each  tenth  in  the  rate  of  change  of  grade. 

This  would  call  for  a  curve  1200  feet  long  in  the  last  example. 

It  is  not  always  practicable  to  meet  this  requirement,  but  such 
curves  could  l:e  and  should  be  much  longer  than  they  are  gener- 
ally made. 


112 


FIELD-MAXUAL    FOR 


ELEVATION  OP  THE  OUTER  RAIL  ON  CURVES. 

A  car  of  weight  w  moving  on  a  curve  of  radius  R  with  a  velocity 
of  v  feet  per  second  develops  a  centrifugal  force  in  the  direction 

ab  (Fig.  66)  expressed  by 


FIG. 


To  counteract  thig  force  the 
outer  rail  on  a  curve  is  raised 
|c    above  the  inner  rail  an  amount 
be  =  e,   so  that   the  car   may 
rest  on  an  inclined  plane. 

Let  ac  =  g,  the  gauge  of  the 
track. 
The  component  of /in  the  direction  of  ac  is 

ab         ab          f          wtf        ab 
f~m=f~g~f    =  32.15J?  '  7* 

The  component  of  w  in  the  direction  of  cq  is 

be  e 

w'  =  w —  =  w—. 

ac          g 

Now/'  and  w'  are  opposite  in  direction,  and  in  order  to  satisfy 
the  mechanical  conditions  they  must  be  equal. 
Equating  their  values,  therefore,  we  find 


32.157? 


(19) 


But  ac  —  distance  between  rail  centers  =  gauge  -f  one  rail 
head  =  4.708  -f-  .202  =  4.91.     For  an  elevation  of  6  inches 

ab  =  ac  —  .03  =  4.88,  nearly. 

This  is  a  good  average  value  for  ab. 

Again,  if  V  =  the  velocity  in  miles  per  hour,  we  have 

_  5280          22 
~  3600          15    ' 


LEVELING,  STADIA    MEASUREMENTS,  ETC.  113 

73 

Furthermore,  R  =  —  -,  in  which  D  is  the  degree  of  a  curve  of 

radius  R,  and  7?i '=  the'radius  of  a  one-degree  curve  =  5729.58. 
Substituting  these  values  in  (19),  we  have 

4.88X484DF* 


bs.io  X  225  X  5729.58 
=  .000057Z)F2,  almost  exactly (20) 

22 

If  we  substitute  in  (19)  ab  =  g,  and  v  =  —  V,  we  have,  approx- 

15 


iniately, 


_  .  06688,7  F2 


This  is  the  formula  in  general  use. 

To  give  the  most  favorable  view  possible  of  this  formula,  s-ub- 
stitute  in  it  g  —  4.708  and  find 


.3149** 
~~ 


Example.  —  Find  the  elevation,  e,  for  a  7°  curve  and  a  velocity 
of  40  miles  per  hour. 
Eq.  (20)  gives 

e  =  .000057  X  7  X  1600  =  .638  feet  -  7.66  inches. 
Eq.  (22)  gives 

«  =  -31*9*1600  =  .616  feet  =  7.39  inches. 
81o.O 

It  will  be  observed  that  the  two  large  factors  appearing  in  (22) 
do  not  occur  in  (20)  ;  and,  moreover,  eq.  (20)  is  very  much  more 
accurate  than  the  formula  (22)  generally  used. 

The  author  has  elsewhere  pointed  out  the  evil  of  elevating  the 
outer  rail  on  sharp  curves  sufficiently  to  meet  the  requirements  of 
too  high  velocities. 

The  elevation  should  be  much  less  than  required  for  the  speed  of 
the  fastest  passenger-coaches  ;  for  it  is  better,  on  such  a  curve, 
for  a  coach  to  hug  the  outer  rail  somewhat  or  to  slacken  speed,  or 
both,  than  to  pull  all  the  freight-cars  against  the  inner  rail. 

The  best  conditions  are  realized  when  the  speeds  of  fast  trains 


114  FIELD-MANUAL    FOB   ENGINEERS. 

are  lessened  so  as  to  exceed  only  slightly  the  speed  for  \vlucli 
the  elevation  of  the  curve  is  suited. 

Thus  a  train  moving  at  a  speed  of  50  miles  per  hour  should  be 
decreased  to  a  35-mile  rate  or  less  in  passing  over  a  curve  elevated 
for  a  30-mile  rate.  Under  such  conditions  the  cars  would  slightly 
press  the  outer  rail  around  the  curve  ;  their  motion  would.thus 
be  steadied,  and  the  movement  would  be  the  steadiest  possible. 

The  maximum  elevation  should  probably  not  exceed  eight 
inches,  except,  possibly,  at  some  special  point,  under  peculiar 
conditions. 


CHAPTER  VI. 


COMPOUND  CURVES. 


A  COMPOUND  curve  consists  of  two  or  more  consecutive  circular 
arcs  of  different  radii;  any  two  adjacent  ones  having  a  common 
tangent  at  their  point  of  meeting,  and  their  centers  on  the  same 
side  of  that  tangent. 

This  common  point  of  tangency  is  called  a  point  of  compound 
curve,  or  P.  C.  C. 

Compound  curves  are  used  to  bring  the  line  of  the  road  upon 
more  favorable  ground  than  any  simple  curve  could  occupy. 

Let  AP  and  PB  (Fig.  67)  be  the  two  branches  rf  a  compound 
curve  uniting  the  two  tangents 
AV&nA  BV.  Continue  APto 
k,  where  the  tangent  kD  is 
parallel  to  BV,  and  draw  the 
chords  AP,  Pk,  and  PB. 

To  prove  that  the  chords  Pk 
and  PB  coincide. 

Since  the  radii  PO,  and  P0a 
are  each  perpendicular  to  the  »< 
common  tangent  at  P,  they  co- 
incide ;  and  since  the  radii  Oik 
and  0.2B  are  perpendicular  to 
parallel  tangents  Dk  and  BV, 
they  are  parallel.  Hence  the 
vertex  angles  of  the  isosceles 
triangles  PO^k  and  PO^B  are 
equal,  and  the  angles  at  the 
bases  must  be  equal  also.  Hence  OiPk  =  O^PB,  and  therefore 
Pk  and  PB  coincide. 

Prolong  Ak  to  meet  BV\n  E.  Let  R,  —  the  first  radius  A0l} 
and  .#3  =  the  second  radius  POiO*.  Let  Oi  =  AO^P,  the  central 

115 


116  FIELD-MANUAL  FOR   ENGINEERS. 

angle,  subtended  by  the  arc  of  radius  .ZJ,,  and  0*  =  FO^B,  the 
central  angle  subtended  by  the  arc  of  radius  7?3. 

Now        AkP(=BkE)  =  £0,,   by  («),  Chapter  IV. 

Also,         kAP  -  PkD  =  PBV  -  \0*  ,  by  (d),  Chapter  IV, 

In  the  triangle  ABVlet  the  tangent  A  V  =  Tlt  and  the  tangent 
BV  =  TI.  We  observe  that  the  shorter  tangent  is  adjacent  to  the 
shorter  radius,  and  the  longer  tangent  is  adjacent  to  the  longer 
radius.  Let  AB  =  c,  BA  V  =  A,  AB  V  =  B. 

.-.  BVF  =  V  =  A  +  B  =  0,  +  0a. 
Draw  kag  parallel  to  A  V. 


aB+ak  _  tan  \(akB  -f  aBk)  _  cot  |<a&#  -  aBk) 
aB-ak~  tan  Ka&i?  -  aBk)  ~~  cot 


See  eq.  (35),  Chapter  III. 

A)=  ^s  +  ^  -  2AX> 


-T4-T  l    • 

ri~colTF' 

aB  -ak-  B  V-  A  V  =  T*  -  T,. 
Also 

aBk  =  DkP  =  |02  ,     and    akB  =  F-  }02. 
Hence 

«A;2?  —  aBk  —  V  -  0*  =  Ol  ,     and    akB  -f  aBk  =  V. 
Substituting  these  values  in  the  above  equation  gives 
T*+Ti  2R,  cot  JO, 


2\  -  rl\        (T9  —  Ti)  cot  \V      cot 
Hence 


' 


i)  cot  IF-  l(T,  -  7\)  cot  40,        ) 

•     •     (1) 
=  Ti  cot  ^F- 


COMPOUND   CURVES.  117 

Similarly, 


y,)  cot  JF- 
cot  £02  =  -         —  ;  =-  -  *— 


or 

^2  =  $(Ta  +  77i)  cot  I  V  —  \(I\  —  Ti)  cot 

L   -  (3) 


(3) 


From  (1)  and  (2)  we  get 

(ft  -  ft)  =a  |(7T2  -  IVXcot  ^0  +  cot 


We  observe  that  either  radius  can  be  found  by  two  multiplica- 
tions and  both  by  three  multiplications,  since  -|(Ja-f-  TI]  cot  \V 
is  common.  This  is  one  multiplication  less  than  is  necessary  by 
any  other  formulas.  The  first  equations  of  groups  (1)  and  (2)  are 
preferable  when  both  radii  are  to  be  found,  but  the  last  equation 
of  either  group  is  best  when  only  one  radius  is  required.  Since 

|F=  £0,  -f  i02  <  90°,     i0j  <  90°,     and    £0a  <  90°, 

Supposing  the  triangle  AB  V  unchanged  ;  then,  since  0j  -j-  0a 
is  constant,  02  varies  inversely  with  0,. 

Eq.  (] )  shows  that  ft  varies  directly  with  Oi ,  and  eq.  (2)  shows 
that  Rv  varies  inversely  with  02  and  therefore  directly  with  0,. 
Hence  the  radii  vary  directly  with  the  central  angle  corresponding 
to  the  shorter  radius.  This  shows  the  advantage  of  making  Oi  as 
large  as  possible. 

Prolong  the  tangent  Dk  to  meet  .Z?02  in  H,  and  draw  the  per- 
pendicular PbL.  Now 

BH  =  BL  —  kb  =  ft  vers  02  —  ft  vers  02  =  (ft  —  ft)  vers  0a<> 

Then  ^  ,   —     .      T_  — 

sin  V  sin  V 


118 


FIELD-MANUAL   FOR   ENGINEERS. 


Hence 
(AD-\-DV)smV  = 

Then    vers  02  = 

and  7?2  •= 

Similarly 


7i  sin  F 

?i  (vers  F  —  vers  02)  ~f-  Ru  vers  02. 

r,  sin  V-R,  vers  F 


i  sin  F  -  /*i  (vers  V-  vers  Oa) 


vers  03 


sin  F=  /A  vers  Ol 


(vers  V  —  vers 


y2  sin  F—  7?3  (vers  F—  vers  0,) 

—  -    - 


or 


sin  V-  R*  vers  F 


It  should  be  noticed  that  the  foregoing  equations  are  symmetrical 
with  reference  to  the  radii,  the  tangents,  and  the  central  angles, 

and  hence  eq.  (2)  may  be 
written  at  once  from  eq.  (1) 
or  nee  versa,  by  simply  ex- 
changing the  radii,  the  tan- 
gents, and  the  central  angles; 
and  the  same  is  true  of  eqs. 
(4)  and  (5). 

Let    Fig.     68    represent    a 
compound    curve,    in    which 
B  A  0,  =  Rt ,  BO*  =  #2 ,  P  =  the 
/  /     P.C.C. 

Drop  the  perpendiculars 
Oiin  and  O-iii  upon  the  chord 
AB,  and  draw  Oi  /^parallel  to 
AB.  Let  I  be  the  intersection 
of  AB  and  P0, ,  and  I  the  an- 
gle AlP.  We  have 

Oyll  —  Oi 02  sin  I  =  (R?  —  Rj)  sin  I. 
Also,         027/  =  02n  —  0,  w  =  722  cos  B  —  Ri  cos  A. 
Equating  these  gives 

-R2  cos  B  —  Ri  cos  A 


COMPOUND   CURVES.  119 

Again,        Am  =  R!  sin  A, 

mn  =  0i#=  0i  0a  cos  2  =  (R»  —  Si)  cos  I, 

nB  =  7?a  sia  5. 
Adding  gives 

(7  =  M!  sin  .1  -f  ^2  sin  B  -f  (.Kg  —  J?j)  cos  £  .     .     .     (6) 

Now  r,  =  o  ^?  ,    and     r.  =  (7  4"-4, 

sm  F  sin  F 

When  72 1  ,  .Ra,  A,  and  Bare  given,  ',  C,  and  the  tangents  J^  and 
Ta  may  be  found  from  the  above  equations. 
We  have 

I  =  0,^  +  AOil  =  (90°  -  .4)  -f  0! ,     or     0,  =  -A  +  J  -  90°. 
Also, 

J  +  J50a  +  10*B  =  180°,     or    I  -f  (90°  -  1?)  +  03  =  180°, 
or  Oi  =  B  -1  +  90°. 

When  /?,  ,  7?2 ,  J.,  and  5  are  given,  we  may  find  I,  0,  ,  and  02 
from  these  equations,  and  then  Ti  and  rl\  from  (4)  and  (5). 

To  find  the  limit  in  one  direction  of  each  radius  of  a  compound 
curve. 

We  observe  that  a  curve  run  from  A  (Fig.  67)  with  a  radius 
I?  =  T7!  cot  ^F  would  be  tangent  to  B  Fat  Et  VE  being  equal 
to  VA. 

A  curve  run  from  A  or  from  any  other  point  of  this  curve  with 
a  radius  greater  than  rl\  cot  \V  would  lie  outside  of  the  above 
curve  and  would  intersect  VE.  Hence  the  smaller  radius  must 
be  less  than  Tt  cot  \  V. 

Again,  a  curve  run  from  B  with  a  radius  11  "=  7T3  cot  \  V  would 
be  tangent  to  VA  prolonged  at  B'  ( VB'  being  equal  to  VB)  and 
would  pass  inside  of  A. 

A  curve  run  from  B  or  from  any  other  point  of  this  curve  with 
a  radius  less  than  Ti  cot  £  F  would  lie  inside  of  the  above  curve 
and  would  pass  still  further  from  A,  Hence  the  greater  radius 
must  be  greater  than  7\  cot  \  F. 

From  trigonometry  we  have 

cot  (45°  -  9)  -f  cot  (45°  -f  2)  =  2  sec  2z  >  2. 


120 


FIELD-MANUAL   FOR   ENGINEERS. 


Therefore 

cot  (45°  -  -a  —  «)  +  cot  [(45  —  a  -f  z)]  >  2  sec  2s  >  2. 

Hence  if  the  sum  of  two  angles  <  90°,  the  sum  of  their  co- 
tangents >  2,  and  the  more  unequal  are  the  angles  the  greater  is 
the  sum.  of  their  cotangents. 

Now  since  V  <  180°,  £0,  -f  40a  =  \V  <  90°. 

Therefore  cot  \Q\-\-  cot  -|0a  >  2  in  all  cases,  and  eq.  (3)  shows 
that 

HI  —  Ri  >  TI  —  TI  in  all  cases (e) 

Again   from  eqs.  (1)  and  (2), 

R*  >  TI  cot  4  V,     and    l?i 


cot 


Therefore 


JR, 


These  equations  show  that  the  radii  differ  more  than  the  tan- 
gents do,  both  relatively  and  absolutely. 

•n     71 

Putting  Ox  =  0a  in  (3)  gives  ' ^  _      '   =  cot  |0X  =  cot  iK. 

The  following  table    shows  the  value  of  this  ratio  for  some 
values  of  V. 

For  the  more  common  values  of  V,  say  40° 
or  less,  we  see  that  the  difference  between 
the  radii  is  several  times  greater  than  the 
difference  between  the  tangents;  and  we  may 
remember  that  when  Oi  and  0a  are  unequal 

7p       Tf 

the  ratios  of  -= ;=-  are   somewhat    larger 

-fa   —    -Li 

than  those  given  in  the  table.  But  the  tabular 
results  are  sufficiently  accurate,  and  they  will 
aid  much  in  assigning  practical  values  to  the 
radii  when  the  tangents  are  known. 

Of  the  quantities  7?,,  Jfa,  0i,  and  02,  and 
the  sides  and  angles  of  the  triangle  ABV,  four  independent 
quantities  must  be  given  in  order  to  find  the  others.  Or  three 
besides  V  must  be  given.  Of  course  the  parts  of  the  triangle 
must  be  consistent,  and  all  of  the  given  quantities  must  be  con- 
sistent with  the  necessary  relations  existing  among  them. 
For  example,  we  must  have  A  -f-  B  =  0j  -j-  0a  =  V, 


COMPOUND   CURVES.  1 

Vis  supposed  to  be  known  in  all  cases.  If  either  Oi  or  02  is 
known,  the  other  is  known  from  the  relation  above. 

PROBLEMS. 

I.  Given  Ti  and  J'2,  also  either  one  of  the  quantities  Ri ,  It?,  0i, 
and  02,  to  find  the  others. 

Example  l.—T*  =  447.32,  rl\  =  510.84,  F  =  15°,  and  R, 
-  3000. 

.-.  jP2  +  T,  =  958.1,     T8  -  1\  -=  63.52,  %(T*  -  T^  =  31.76, 

and  cot  £F=  7.596. 

(1)  gives 
cot  j0l  =  mix™* -6000  =  787764-6000  =  2fl ^ 

Do.  O^;  uo.O^ 

and  |0,  =  2°  50'  45". 

Now    ^02  -  %V  -  i-Oi  =  4°  39'  15",     and     cot  i02  =  12.283. 

.-.  cot  £0i  -f  cot  £02  =  32.398. 
From  (3),     R,  =  31.76  X  32.398  +  3000  =  4028.96. 

If  7?2  is  given,  find  in  the  same  way  £02,  £0i,  and  7?!. 
If  0!  is  given,  03  =  F-  0,. 
Then  find  #1  from  (1),  and  R*  from  (2). 
Similarly  if  02  is  given. 

Example  2  (see  Henck,  Prob.  48).— Let  7\  —  480,  T*  —  500, 
F=18°,  and  01=£F=9°. 

.-.    02  =  9°. 

In  this  case  the  common  tangent  at  the  P.C.C.  makes  equal 
angles  with  the  tangents  A  Fand  BV.  We  have 

KZ;  +  i\)  =  490,   i(rt  -  ro  =  10, 

cot  £F-  6.31375,  cot  £0x  =  cot  -i09  =  12.7062. 
(1)  gives 

Rl  =  490  X  6.31375  -  10  X  12.7062  =  2966.67. 
(3)  gives 

R,  =  10(12.7062  +  12.7062)  -f  2966.67  =  3220.79. 


FIELD-MANUAL   FOR   ENGINEERS. 

II.  Given  A,  B,  and  (7  and  any  one  of  the  quantities  jR,  ,  7?2  , 
0,  ,  and  03  ,  to  find  the  others. 

Example  1.—  Let  A  =  8°,  B  =  7°,  and  C7=  950.     Let  R,  =  3000. 
We  have 


sin  V 

Ta  =   ^y  =  8670.5  X  13917   =  510.84. 
Eq.  (1)  gives 


.12187  =  3670.5  X  .12187  =  447.32; 


_  958.16  X  7.59575  -  6000  _  1277.94 

CO    fro  ~ /'«>    K0~     == 

63.52  03.52 

.«.  40,  =  2°  50'  45";     .  %  ^0,  =  4°  39'  15",    and     cot  £0a  =  12.2836. 
Eq.  (3)  gives 

&  =  T*~  Ti  (cot  |0r+  cot  |0a)  +  Ifc 
=  31.76  X  32.4024  +  3000  =  4028.96. 

If  Hi  is  given,  find  T\  and  T*  as  above,  and  then  02 ,  Oj  ,  and 
Jib,  in  order. 

If  0,  is  given,  we  have  02  =  (A  -f  #)  -  0,. 

Find  Ti  and  T2  as  shown,  then  ^  and  /?,  from  (1)  and  (2). 
Similarly  if  02  is  gi\en. 

Example  2.— Let  Oi  =  A  =  %°\  then  0*=  B  =  7°,  and  (7  —  950. 
Let  ,R  =  3000. 

In  this  case  the  common  tangent  at  the  P.C.C.  is  parallel  to  the 
chord  AB.  Ti  and  Ti  are  found  as  above.  Then  eq.  (1)  gives 

Si  =  K^  +  T,)  cot  ^F-  £(T9  -  TO  cot  -1(9, 
=  479.08  X  7.59575  -  31.76  X  14.3007 
=  3638.97  -  454.19  =  3184  8, 

and         #a  =  -~^  (cot  £0,  -f-  cot  1  Oa)  +  5, 

=  31.76(14.3007  -f  16.35)  +  3184.8  =  4158.26. 


COMPOUND   CURVES.  1^3 

One  of  the  most  useful  problems  in  compound  curves  is  that  of 
fitting  such  a  curve  to  tlie  ground.  The  lengths  of  the  branches 
give  the  corresponding  central  angles,  the  sum  of  which  is  equal 
to  the  deflection  angle  V. 

We  select  a  problem  proposed  by  "  L.  E.  Vel  "  in  Engineering 
News,  March  5,  1881. 

III.  "Two  tangents  A  V  and  BV  intersect  at  an  angle  of  60°. 
It  is  required  to  begin  at  the  point  A  and  locate  2000  feet  of  2° 
curve,  followed  by  2000  feet  of  1°  curve. 

"  Required  the  distance  to  bo  measured  back  from  the  intersec- 
tion V  to  the  point  of  curve  A  by  the  shortest  and  simplest 
method." 

A  solution  of  this  problem,  marvelous  for  length  and  labor 
involved,  can  be  seen  in  Engineering  News  of  March  19,  1881. 
(See  also  Searh  s  Field  Engineering,  Art.  164.) 

Being  challenged  by  the  contriver  of  tlie  above  solution,  the 
author  gave,  in  Engineering  News  of  April  2;  1881,  in  substance 
the  following  as  the  "shortest  and  simplest"  solution  possible: 

We  have          R*  =  2JB,,     or    7?2  -  R,  =  /?,. 

Substituting  Rl  for  #2  —  Rl  in  (4),  observing  that  02  =  20°, 
we  have 

T,  =  R,  (vers  20°  -f  vers  60°)  •£  sin  60° 

=  2864.8  X  .56031  -f-  86603  =  1853.5.         Q.E.D. 

If,  for  any  reason,  T2  is  desired,  we  have,  from  (5), 

_  ZRi^^y—  &  vers  0»  _  Ri(i  —  vers  0»)  _  Ri cos  0i 

"sliTF  sin  V  sin  V~ 

-  2864.8  X  .76604  -4-  .86603  =  2534.04. 

IV.  (liven  Rl   and  7?2  and  either  tangent,    to  find   the   other 
tangent  and  the  central  angles  Oi  and  02. 

Example.— Let  V  -  72°,  3\  =1091,  Rl  —954.9,  and  R*  — 
3437.7.  sin  V  =  sin  72°  =  .95106,  vers  V  =  .69098,  and 
R,  -  R,  =  2482.8. 

(4)  gives 

vers  0,  * 

.-.     Oa  =  32°  1'  20"     and     0,  =  F-  Oa  =  39°  58'  40"; 
vers  0,  -  .23371. 


124  FIELD-MANUAL   FOR 

Now  (5)  gives 

_  954. 9  X. 23371+  3437.7  X  .45727 
.95106 

V.  Given  either  tangent  and  the  adjacent  or  the  opposite  radius, 
also  either  central  angle  0,   or  02 ,  to  find  the  other  tangent  and 
radius. 

Example  1.— Given  AV '=  T>  =  1091,  R,  =  954.9,  V=  72°,  and 
0,  =  39°  58'  20".  Hence  02  =  32°  1'  40". 

(4)  gives 

1091  X  .95106  -  954  9(.69098  -  .15216) 
.15216 

Then  T*  is  found  from  (5)  as  above. 

Example  2.— Given  2\  =  1091,  #2  =  3437.7,  and  the  angles  as 
above. 
From  (4), 

n  _  1091  X  .95106  -  3437.7  X. 15216  _ 
.53882    " 

Then  TV  is  found  from  (5)  as  above. 

VI.  Given  the  tangents  and  a  central  angle   Oi  or  02,  to  find 
the  radii. 

Iti  and  7?2  are  found  from  (1)  and  (2). 

Example.— Lei  Ti  =  480,  I\  —  500,  V—  18°  and  Oi  =  9°. 

Then  02  =  18°  -  9°  =  9°. 

Now        cot  \V—  6.31375  ;  cot  \0,  =  cot  ^02  =  12.7062. 
Now,  from  (1), 

Rl  =  240  X  19.02  -  250  X  6.39245  =  2966.7  ; 
and  from  (2), 

Rv  -  250  X  19.02  -  240  X  6.39245  =  3220.8. 

SPECIAL  PROBLEMS. 

It  frequently  happens  that  a  curve  already  located  must  be 
changed  to  meet  certain  conditions.  In  this  case  the  equation 
must  be  solved  anew  or  the  results  already  obtained  must  be 
modified. 


COMPOUND    CURVES. 


125 


If  the  required  changes  are  small,  and  especially  if  some  latitude 
is  allowed  in  making  them,  the  solution  can  sometimes  be  readily 
modified  to  conform  to  the  new  conditions  ;  but  it  is  generally  best 
to  solve  the  equation  anew. 

This  will  be  illustrated  in  the  following  problems. 

Having  located  a  simple  curve  APB,  radius  A0\  =  Ri  ,  ending 
in  the  tangent  BV,  it  is  required 
to  endwithagiven  shorter  radius, 
B'O'  =  It*,  in  a  tangent  RV\ 
parallel  to  BV,  and  at  a  distance 
within  BV  equal  to  BH  =  p. 

The  distance  BP  back  to  the 
P.C.C.,  and  the  distance  HB'=d, 
that  the  end  of  the  new  curve 
is  back  of  the  old,  are  required. 

Let  PGfB  =  P01B=  0. 

Now 


cos  0  = 


KO,       BO,-HK-BH 


0,0' 


0,0' 


fa  -fa 


vers  0  = 


z>  z_>  * 

Zfcl      JtfQ 


—, 
vers  0 


P 
vers  0" 


(7) 


Or  proceed  as  follows:   Describe  the  arc  0' C with  center  0lt 
Now 

CK 


vers  0  = 


0,0'' 


But          CK  =.  BK  -  BC  =  BK  -  PO'  =  BK  -  fa  , 
and  BH  =  BK  -  HK  =  BK  -  fa. 


=  BH=p. 


Hence,  as  before, 


0  = 


Also, 


d  =  Hff  =  KO'  =  (R,  -  St)  sin  0. 


(8) 


126  FIELD-MANUAL    FOR   ENGINEERS. 

From  (7)  and  (8), 

d        sin  0 


p       v«rs  0 
d  =  p  cot  | 


=  cot  $0, 


(9) 


Equation  (9)  is  easily  proved  as  follows  :  Draw  PB'B,  which  we 
have  seen  is  a  straight  line.     We  have 


BB'H  =  B'BV  =  \PO,B  =  10. 
Hence  HB'  =  d  =  HB  cot  BB'H  =  p  cot  ^0. 

Example  L— Having  located  a  3°  curve  ^ PI?  (Fig.  69),  radius 
1909.9,  required  the  distance  back  from  B  to  the  beginning  of  a 
5°  curve  (radius  1145.9)  that  will  end  in  a  tangent  27  feet  within 
the  tangent  BV,  also  the  distance  HB'. 

(7)  gives 

vers  (PO'B'  =  PO,B)  =  ,y?  =  .03534; 
.-.    0  =  15°  17'. 

Then       15    *'    =  5.09  chains  =  509  feet  =  arc  BP. 
3 

Also,  JIB'  —  p  cot  %0  .-=  27  X  7.453  =  201.2  feet. 

II.  Suppose  the  curve  APB  is  located,  and  it  is  desired  to  end 

with  a  longer  radius,  PO'  =  R?, 
on  a  parallel  tangent  B  V' ,  at  a 
V        ,    distance  HB  =  p  without  BV. 

Similar  equations  to  the  above 
apply. 

Let  7*0,5 =PO'B'=0.  Then, 
as  in  Fig.  09, 

p  =  BI1  =  B'K  -  BD 
—  Rt  vers  0  —  Hi  vers  0 

=  (R9  -J^)  vers  0.    »    (!') 
Also,  d  =  RH=KD 

=  (R*  -  Jv'Osin  0,  (8') 
and        d  =pcottO.    .     .     (9') 


COMPOUND   CURVES.  127 

Example  2.  —  Having  located  a  5°  curve  APS,  required  the  dis- 
tance back  from  B  to  the  beginning  of  a  3°  curve  that  will  end 
in  a  tangent  27  feet  without  BV. 

PO,B  =  PO'B'  =  0. 
We  have 

vers  0  =  TVr  =  .03534. 
.-.   0-15°  17'. 

Then         155<>1  '    =  3.06  chains  =  306  feet  =  arc  BP. 
Also,  B'H  —  p  cot  $0  =  27  X  7.453  =  201.2  feet, 

the  distance  that  the  new  tangent  point  B'  is  in  advance  of  the 
original  tangent  point  B. 

With  reference  to  the  preceding  problems,  we  observe  that  if 
any  two  of  the  following  quantities  :  the  new  radius,  the  offset 
,  the  tangent  distance  HB'  =  (1,  the  arc  BP  (or  BP)  or  the 
equivalent,  the  angle  0,  are  given,  the  other  two  may  be  readily 
found  by  the  use  of  the  preceding'  equations. 

Example  3.  —  Let  p  =  1,  and  let  PB  =  150  feet,  the  degree  of 
the  curve  APB  (Fig.  69)  being  7°  10'.  Hence 

0  =  10°  45',     Bl  =  800. 


Eq.  (7)  gives       &  =  800  -  —         =  800  -  57  =  743. 

.  Ul  t  OO 

Also,  HB'  =  cot  5°  22£'  =  10.63. 

Example  4-  —  Having  run  two  given  curves,  PB  and  PB'  (Fig. 
69),  subtending  equal  central  angles,  PO,B  =  PO'B'  =  0,  to 
find  the  distance  between  the  terminal  tangents  BV&ud  B'V  and 
the  distance  RE'. 

(7)  gives 

p  =  (#!  —  R.z)  vers  0. 

Then,  from  (8),  we  have 

SB*  =  (#1  -  7»»a)  sin  0. 


128 


Fl ELD-MANUAL   FOR   ENGINEERS. 


These  equations  and  Figs.  69  and  70  snow  that  if  n  curve  is 
sharpened  or  flattened,  so  as  to  end  on  a  tangent  parallel  to  the 
original  tangent,  the  new  curve  does  not  end  at  a  point  opposite 
to  the  original,  but  is  moved  backward  or  forward  according  as 
the  curve  is  sharpened  or  flattened. 

Problem.  —  Given  a  compound  curve  APE'  (Fig.  71)  ending  in 
the  tangent  B'V,  the  first  radius,  A0l  =  POi  =  Rt,  being 

shorter  than  the  second 
radius,  P0*=  R*\  to  change 
the  P.C.C.  and  also  the 
last  radius,  so  that  the  new 
curve  radius  OP'  =  1? 
shall  end  in  a  given  par- 
allel tangent  B"V". 

It  is  evident  that  the  new 
P.C.C.  must  be  on  the  first 
branch  AP,  since  if  on  the 
second  branch  the  problem 
would  be  the  same  as  the 
FIG.  71.  preceding.  The  P.C.C.  (at 

P')  must  be  found  so  that  the  arc  P'B'  will  be  tangent  to  B"  V"  . 
Suppose  the  first  branch  AP  of  the  curve  extended  to  B,  where 
tangent  BV  is  parallel  to  B'V. 

Let    A01P=01,     PO*B'  =  03,     and    P'OB"  =  0. 


Let    B'H=a, 


B"C  =  p,    BH  =  d,     BK=d'. 


I.  Suppose  R'  <  Ri  <  Ri.     Then  B"V"  is  necessarily  with- 
in B'V. 

From  the  preceding  problem  we  have  directly 

B'H  =  (B,  -  Ri)  vers  02  =  a;   .......     (10) 

BH=  OjOa  sin  0,  =  (R9  -  R,)  sin  02  =  d.     .     .     (11) 


Dividing  gives 
vers  0a 


—  _,     or    d  =  a  cot  £02.  .     .     (12) 
d 


COMPOUND    CURVES.  129 

a  and  d  are  expressed 'in  terms  of  known  quantities,  and  are 
always  known. 
Again, 

B"K=  (#,  -  .R')  vers  0  =  a',     or    vers  0  =  _    a        •   .     (13) 

Mi  —  H 

BK  —  00l  sin  0  =  (#,  —  R')  sin  0  =  d' (14) 


.-.  tan  \0  —  a-,,     or    d'  =  a'  cot  ^0.     ...     (15) 


Since  the  position  of  the  tangent  is  given,  a'  is  known  and  eq. 
(13)  determines 'the  beginning  of  the  curve.     Also, 

HK  =  d  +  c V  —  a  cot  ^02  -f  a'  cot  £0,      .     .     (16) 

which  determines  the  end  of  the  curve. 

2.  If  the  curve  is  to  begin  at  a  certain  point,  0  is  known,  and 
we  have,  from  the  above  eq.  (13), 

radius  =  11'  =  ^ --'—-. 

versO 

Then  the  end  of  the  curve  is  found  as  above. 

3.  If  the  curve  is  to  end  at  a  given  distance  back  (or  front)  of 
13',  d'  is  known,  and  we  have 

tan  10  =  ~, 

which  determines  the  beginning  of  the  curve. 
Then  the  radius,  R',  is  given  as  above. 

II.  Suppose  R'  >  2?,.     (Fig.  72.) 

(a)  Let  B"V"  be  within  B'  V.     Then  R  <  7?2. 

We  have           B'II=  (R*  -  Rt)  vers  0,  =  a;    ....     (10') 
BH=  (R-2  -  A'Osin  Oa  =  d (11') 

.-.     tail  40,  =  ' (12') 


130 

Also, 

and 


FIELD-MANUAL   FOIi   ENGINEERS. 

B"K=  (R'  -  R,)  vers  0  =  a';  .     .     .     .     (13') 
BK=  (R1  -  #0  sin  0  =  d';    .     .     .     .     (14') 


tan  10  =  -. 
d 


(15') 


FIG.  72. 


Also> 


Finally, 


HK  =  £#  -  BK  =  d  -  d1. 


(16') 


2.  If  tlie  beginning  of  the  curve  is  fixed,    0  is  known   and  the 
radius 

a' 


=  R'  =  R1  -f 


vers  0' 
HK  =  d  -  d'  =  a  cot  |0a  —  a  cot  £0, 

giving  end  of  the  curve. 

3.  If  the  end  of  the  curve  is  tixed,  d'  is  known  and 


_ 

d'' 


Then 


R'  =  R, 


vers  0' 
(1))  Suppose  B"  V"  is  without  E'  V.     (Fig.  73.) 


COMPOUND   CURVES. 
In  this  case  we  may  have 

R'  =  JBa. 
> 

We  have,  as  already  explained, 

B'H  =  (#„  -  Et)  vers  Oa  =  a,       .     . 
BH  =  (R*  -  R,)  sin  02  =  d, 


and 


C          B 


FIG.  73. 


Also, 
and 


^  =  (R'  _  ^)  Vers  0  -  «', 
BK  =  (R'  -  ft)  sin  0  =  d', 


and 

By  transposition, 


tan  $0  =  -,. 


vers  0  =  -,- 


R'  -  RS 

which  determines  the  beginning  of  the  curve. 
By  subtraction, 

HK=d'-d,      . 
which  determines  the  end  of  the  curve. 


132  FIELD-MANUAL    FOE    ENGINEERS. 

A  special  case  of  this  problem  is  that  in  which  the  P.  ('.  C.  only 
is  changed.     Then  11'  =  #2,  as  illustrated  in  Fig.  73. 
B''V"  is  without  B'V.     (Fig.  73.) 
(a)  Substituting  R^  for  R'  in  the  value  of  a' ,  we  have 

of  —  a  =  p  =  (R.2  —  J?i)  (vers  0  —  vers  03). 
.'.     vers  0  =  vers  02  -{- — — -. 

Then  #2  =  Rt 


vers  0' 

P'OiP  =0—0?  shows  the  angular  distance  that  P'  is  back 
of  P. 

(b)  We  may  consider  that  the  curve  AP*B"  is  first  located,  and 
that  APB'  is  the  new  curve,  the  new  tangent  B'  V  being  within 
B"  V".  Then 

P'OB"  =  03,     and     PO*B'  =  0, 
and  we  have 

7) 

vers  0  =  vers  02  —  — — - — =-. 


a 

=  #1  4          -7;,  as  before. 
vers  0 


P'OiP  shows  the  angular  distance  that  the  new  P.C.C.  (P)  is 
in  advance  of  P'  . 

2.  If,  in  this  problem,  the  beginning  of  the  curve  is  fixed,  0  is 
known  and  we  have 

S  =  ft  +  —  , 
vers  0 

The  end  of  the  curve  is  given  by 

UK  =  d'  -  d  =  ^'  cot  ^0  -  a  cot  ^02. 

3.  If  the  end  of  the  curve  is  fixed,  d'  is  known  and 

tan  10  =  ^,. 
This  determines  the  beginning  of  the  curve, 


COM  PO  U  N  D    C  r  R  V  ES. 


13; 


Then  the  radius  —  R'  — 

4.  A  special  case  of 
the  general  problem  is 
that  in  which  the  new 
curve  ends  on  the  same 
radial  line  as  the  original 
curve. 

(a)  OB"  coincides  with 
OiB' ',  and  //  with  K, 
as  shown  in  the  figure. 
Hence 

BK=BH,  ord'  =  d. 

We   have    from    equa- 


vers  0 


H     B'  B" 


FIG.  74. 


tions  preceding,  by  subtracting  and  transposing, 


tan  10  —  tan  \0^  -f  — ,  —  ^  =  tan  ^02  +  ^— ~   =  tan  i02  +  ^. 
d        d  d  a 


This  gives  the  beginning  of  the  curve.     We  have 

0}L  =  OiO*  sin  OtOiL  —0^0  sin  O^OL, 
or  (It*  -  Hi)  sin  02  =  (I?  -  2^)  sin  0. 


E'  = 


—      t    sn 


sin  0 


P'OtP=  0  —  Oi  shows  the  angular  distance  that  P'  is  buck  of 
P. 

(b)  We  may  suppose  the  curve  AP'B"  first  located,  and  thai, 
APB'  is  the  new  curve,  the  new  tangent  B'  V  being  within 
B"V".  Then 


"=0^     PO^E'-O. 


tan  \0  —  tan  iO,  -  ~, 


134  FIELD-MANUAL   FOR   ENGINEERS. 


and 


sin  0 


as  before  found. 

In  this  case  the  new  P.  C.  C.  (P)  is  on  the  arc  AP1  prolonged. 
From  the  above  value  of  Ri   we  have 

sin  0    _  R*  —  R, 
sin  03        R'  —  RI' 

which  shows  that  for  0  and  03,  each  less  than  90°,  we  have 
0  =  02  accordingly  as  7?2  =  R'. 

Hence  the  radius  and  the  P.  C.  C.  must  both  be  changed. 

EXAMPLES. 

1.  A  10°  4'  curve,  AP,  is  followed  by  100  feet  of  7°  44'  curve, 
PB'.  It  is  required  to  find  the  beginning,  the  end,  and  the  length 
of  a  17°  48'  curve  that  will  end  in  a  parallel  tangent,  B"VJ',  58 
feet  inside  of  B'  V.  (Fig.  71.) 

We  Lave 

R,  =  569.2,        R,  =  740.9,     and     R'  =  321.9; 
also,  0a  =  7°  44',     and    p  =  58  feet. 

Hence,  eq.  (10), 

B'H  =  (It*  -  7*,)  vers  03  =  171.7  X  .00909  =  1.56  =  a, 
and,  eq.  (13), 

B"K  =  58  -  1.56  =  56.44  =  a', 
and  therefore 

vers  0  =  ll^l  =  .22822.       .-.     0  =  39°  29'. 
Now 

39°  29'  -  7°  44'  =  31°  45',  and  100^-^'  =  315.4  ft. 
Hence  the  beginning  of  the  new  curve  is  315.4  ft.  back  of  P. 


COMPOUND   CURVES. 


135 


Also,  eqs.  (11)  and  (14), 

HK  =  (B9  -  7?,)  sin  03  -f  (Rt  -  R)  sin  0 

=  171.7  X  -1346  -f  247.3  x  .6358  =  180.3  -f, 
or  HK  =    1.56  X  H.79  -f  56.44  X  2.788  =  180.4. 

The  length  of  P'B"  is  100?^-|^  =  221.8  feet. 

2.  Suppose  the  original  curve  and  offsets  a  and  a'  as  above,  and 
that  the  new  curve  is  to  begin  at  P',  315.4  feet  back  of  P.  To 
find  It'  and  the  end  of  the  curve.  We  have 

P'O^P-  10°  04'  X  3.154  =  31°  45'. 
.-.     FOB"  =  7°  44'  +  31°  45'  =  39°  29'  =  0. 
Then,  eq.  (13), 


R'  =  R,  - 


vers  0 


=  569.2  - 


.22822 


=  321.9. 


The  distance  HK  is  now  found  as  above. 

3.   Suppose  that  it  is  required  to  end  the  curve  a  given  distance, 
HK,  back  of  B'.     Then 

d'  =  UK  —  d      tan    0  =      ;     and     #'  = 


vers  0' 


Problem. — Given  a  com- 
pound curve,  APS',  ending 
in  the  tangent  B'  V ,  the 
first  radius,  AOl  =  Ri)  being 
longer  than  the  second 
radius,  P02  —  Hi,  to  change 
the  P.  C.  C.,  and  also  the 
last  radius,  so  that  the  new 
curve  shall  end  in  a  given 
parallel  tangent  B"  V". 

I.  Suppose  R'  >  Ri  >Rt. 
Then  B"  V"  is  without.B'F7. 
Let  the  notation  be  as  in  the  (5 
preceding  problems.     Then 


FIG.  To. 


136  FIELD-MANUAL    FOR   ENGINEERS. 

B'H=  (Ri  -  Ri)  vers  03  =  a;    ......     (10a) 

BH=  0,0,  sin  03  =  (Rl  -  1?3)  sin  03  =  d.     .     (11«) 

a  and  d  are   expressed   in  terms  of   known  quantities   and   are 
always  known. 
Dividing  gives 


sm  O2 
Similarly, 


°*  =  tan  A02  =  ff-     or     d  =  a  tan  40,. 


(12«) 


B"K  =  (R'  -  R}  )  vers  0  =  «',     or      vers  0  =  _.        ,  (13a) 

ji  —  jii 

^^=  OOj  sin  0  =  (72'  -  J?,)  sin  0  =  ^',       ....     (14a) 
and  tan  |0  =  —  ,     or     rf'  =  a'  cotan  £0i.       .     .     (15«) 

Since  tbe  position  of  tLe  tangent  is  given,  a'  is  known,  and  then 
vers  0  determines  the  position  of  P'  or  the  beginning  of  the  new 
curve. 

Again, 

HK  =  d  4  d'  =  a  cot  iOa  +  a'  cot  {0.      .     .     (16a) 

This  determines  the  end  of  the  curve. 

2.  If  the  beginning  of  the  curve  is  assigned,  0  is  known  and 
we  have 


J.«*      —     -*-*l  s\9 

vers  0 

The  end  of  the  curve  is  given  as  above. 

8.  If  the  curve  is  to  end  at  a  given  distance  in  front  of  B',  d' 
known  and  we  have 

tan  \0  =  -,. 

This  determines  the  beginning  of  the  curve. 
Then  the  radius  =  R'  =  /?,  -)- 


COMPOUND    CURVES. 


13r 


IT.  Suppose  R'  <  Ri. 

(a)  Let   the    terminal    tan-          ^ 
gent  B"  V"  be  without  B'  V , 
then  R'  >  R*. 

Now 


=(U,-R^  vers  0a=a;  (106) 
BH—0^0-,  sin  Oa 


tan  i0,  =      , 

or         d  =  a  cot  £0a.  .     (126) 
Again, 


=i-        vers     = 


'    Oi 


or        vers  U— 


77      (136) 


FIG.  76. 


Ri-R' 

and  BK=  00,  sin  0  -  (JR,  -  J2')  sin  0  =  d';    .     .     (146) 


tan  |0  =  —      or     d'  =  a'  cot  |0.      ...     (156) 
a 


vers  0  = 


determines  the  beginning  of  the  curve,  and 

HK=  BII-  BK^d-  d'  =  a  cot  £02  -  a'  cot  £0      (166) 

determines  the  end  of  the  curve. 

2.  If  the  beginning  of  the  curve  is  given,  find  R'  ^=  Mi  —  —=, 

vers  0 

as  before. 
The  end  of  the  curve  is  given  by  UK  =  d  —  d',  as  above. 

3.  If  the  end  of  the  curve  is  given,  d'  is  known,  and  then 


tan  \0  =  ~. 


This  determines  the  beginning  of  the  curve,  i 
Then  the  radius  is  given  as  above. 


138 


FIELD-MANUAL   FOR   ENGINEERS. 


(b)  Suppose  B"V"  to 'be  within  B'V.     We  may  have 
< 

Now  B'H—  (Ri  —  R*)  vers  0a  —  a;   .     .     .     .     (We) 

BH=  0i02  sin  02 

' — W-Y     =« 

/P'     ^ 

tan  |0a  =  (Lt 


"J5r  =  (#!  -  /*')  vers 0=0', 
K   or       vers  0  = ^__.  (13c) 

^T=  (90,  sin  0 
i?/)sinO  =  d';  (14c) 


FIG.  77. 


or 


d'  =  a'  co 


(15c) 


The  position  of  the  tangent  gives  af,  and  then  eq.  (13r)  deter- 
mines the  beginning  of  the  new  curve. 

HK  ~  d'  -  d  =  a'  cot  \0  -  a  cot  £02      .     .     (16c) 

determines  the  end  of  the  curve. 

1.   A  special  case  of  the  above  is  that  in  which  R'  =  fa.     Sub- 
stituting /?2  for  R'  in  the  values  of  ft'  and  a,  and  subtracting,  we 


have 


CB"  =  p  =  a'  —  a  =  (fit  -  R9)  (vers  0  -  vers  02). 


.'.     vers  0  -  vers  Oa  -f 


and 


--^. 
vers  0 


COMPOUND    CURVES. 


139 


We  may  regard  AP'B"  as  the  original  curve,  APE'  as  the  new 
curve,  the  new  tangent  B'V  being  without  B"V".     Then 


and 


P-OB"  =  02, 

vers  0  =  vers  02 


=  0; 
P 


vers  0' 

as  before, 

2.   If  in  this  problem  the  beginning  of  the  curve  is  fixed,  0  is 
known  and 

R'  =   R,~ 


vers  0' 

The  end  of  the  curve  is  given  as  above. 

3.  If  the  end  of  the  curve  is  fixed,  d'  is  known  and 


tan  £0  =      . 

Then  the  radius  is  found  as  above. 

4.  A  special  case  of  the  last  is  that  in  which  the  new  curve 
ends  on  the  same  radial  line 
as  the  original  curve. 

(a)  OB"  coincides  with 
OiB'  ',  and  .fiTwith  K.  Hence 
d'  =  d.  Then 

tan  0  —  —  7  =  —  ;  tan  Oa  =  —  . 
d        d  d 


.'.     tan  0  —  tan  02  = 


d    '' 


or 

tan  40  =  tan 


+ 


=  tan  |0,  +  J-.  0, 

FIG.  78. 
This  gives  the  beginning  of  the  curve.     Now 

BII=OiOs  sin 02  —  0a< 


0 =(#,-#„)  sin  0^(R,-R')  sin  0. 


140  FIELD-MANUAL   FOU   ENGINEERS. 

From  this  we  have 

(R,  —  R-,}  sin  0o 


R'  = 


(b)  We  may  regard  AP'B"  as  the  original  curve,  and  APB'  as 
the  new  curve,  the  targent  B'  V  being  without  B"  V" '.     Then 


P'OB"  = 


PO,B'  =  0. 


tan  W  =  tan  £0a  —     -, 


and 


„,       w        (Bt  -  R2)  sin  02 
H    =  R  i  —  — ; — ^r —  —  . 

sin  0 


In  laying  out  compound  curves  it  is  generally  best  to  lay  out 
each  branch  from  the  station  at  the  beginning  of  the  branch,  and 
subsequent  stations  if  necessary. 

Thus,  in  Fig.  67,  AP  would  be  laid  out  from  A,  and  other  sta- 
tions on  AP  if  necessary.  Then  the  transit  would  be  moved  to 
P,  directed  along  the  tangent  at  that  point,  and  then  deflected  to 
stations  on  PB,  etc. 

We  will  show  further  on,  however,  how  to  lay  out  one  branch 
of  a  compound  curve  from  a  station  on  another  branch. 

Problem.  —  Given  a  compound  curve  APB,  ending  in  a  tangent 

BV,  to  change  the  curve  so 
as  to  end  on  a  new  tangent 
BV,  but  at  the  same  tangent 
point  as  before. 

I.  Suppose  the  shorter 
radius  to  precede  the  longer. 
(Fig.  79.) 

Let  B  =  the   known    angle 

VBV',AV  =  Ti,BV=  fl\. 

LetAV'=  Tf,  BV'  =  T" 
and  VV  =  a. 

We  have,  from  the  figure, 

V=V-B. 


F'°-79- 


COMPOUND   CURVES. 
In  the  triangle  BVV  we  have 
T"         sin  V 


Also, 


T*         sin  7" 

_o_    _    sin  B 
1\    ~  sin  V" 


in  7'' 


141 


Then 


r  =  ri\  -  a. 


With  these  values  of  V,  T' ,  and  T"  the  tangents  may  be  con- 
nected by  assuming  a  value  for  one  of  the  quantities  Oi,  02,  I\\, 
or  Ri,  and  using  eq.  (1)  or  (2)  and  (3). 

When  V  is  on  the  prolongation  of  AV,  we  have 

V  =  7  +  B,     and     2"  =  Ti  +  a. 

Otherwise  the  solution  is  just  the  same  as  the  above. 

'II.   Suppose   the    longer    radius    to 
precede  the  shorter.     (Fig.  80.) 

We  have  A 

AV=Tlt    BV=T*.    AV'=Tf, 

and  BV  =  T". 

Also,  VV  =  a, 

and,  in  the  figure,  the  angle  V 
=  V+  B. 
„  sin  7 


Then 


Also, 


= 


_       sia  B 

^ 


T'  =  T,  +  a. 


FIG.  80. 


With  these  values  of  V,  T',  and  T"  a  compound  curve  may  be 
located,  as  already  explained. 

Remark. — A  special  solution  of  this  problem  in  a  voluminous 
work  on  Field  Engineering,  with  an  example  worked  out,  occupies 
six  pages. 


14-3 


FIELD-MANUAL    FOR    KXO1NKKUS. 


Problem. — Having  found  the  radius  AO  =  R  of  a  curve,  to 
substitute  for  it  two  radii,  AOi  —  Ri, 
and  P,02  =  7?a,  the  longer  of  which, 
-40i,  is  to  be  used  for  a  certain  dis- 
tance only,  at  each  end  of  the  curve. 


FIG.  81. 


Let      A 
and        Pi 


=  BCP,  =  0,, 


Supposing  the  arc  AP\  described 
with  the  longer  radius,  the  intersection 
of  P,0,  with  the  central  radius  MO  of 
the  simple  curve  in  0*  gives  02  as  the 
center,  and  Pi09  as  the  radius,  of  the 
central  branch  of  the  compound  curve. 

In  the  triangle  00,0.., 


0,003  =  180°  - 


00,02  =  0,, 


and        00,0,  =P10*M=  \V -  0, ; 


00,  =  Si  -  R,     and     0,02  =  &  -  R,. 


Hence 


0.0, 
0,0 


sin  \V 


'R,  -  R    ~  sin  (f7-  0,)' 


(17) 


From  this  we  find 


sin  I V 


'sin  ('7-  0,)' 


(18) 


We  may  assume  values  for  any  two  of  the  quantities  lilf  lit, 
and  0i,  and  find  the  other  from  either  of  the  above  equa- 
tions. 

We  observe  that  0,  -f  0a  =  £  7,  which  is  known.  Hence  if 
a  value  of  0i  is  assumed,  or  if  0,  is  computed  from  Hi  and  ll», 
02  also  becomes  known.  There  are  then  but  three  independent 
quantities,  I?,,  R*,  and  0,  (or  02). 


COMPOUND    CURVES. 


143 


When  this  is  used  as  a  transition  curve,  R:  is  much  greater 
than  It;  the  first  branch  is  short,  or  Oj.  is  small;  hence 


R,  -  R  ~  sin(iF-Oi) 

does  not  much  exceed  unity ;  or  Hi  —  J?2  does  not  much  exceed 
Ri  —  R,  or  JBa  is  not  much  less  than  R.  Hence,  by  sharpening 
the  main  portion  of  the  curve  a  small  amount,  the  ends  of  the 
curve  may  be  much  flattened. 

To  determine  the  distance  MM'  between  the  middle  points  of  a 
simple  curve  and  a  three-centered  compound  curve  joining  the 
same  tangent  points.  (Fig.  81.) 

From  the  triangle  OO^O*, 


Then 


MM'  =  00?  +  0*M'  -  OM 


sin 


(19) 


Problem. — To  connect  two 
tangents  by  a  compound 
curve  of  four  branches,  the 
first  two  consuming  one  half 
of  the  vertex  angle  V. 

Let  A'PitnPiB'  represent 
the  curve. 

Let     A'Ot  =  #,  , 

P,0a  =  R,. 

The  vertex  distance 

Vm  =  E, 
arid       A'OiPi  =  Oi. 

Draw  0ZH  and  0*K  per- 
pendicular to  A'V  and  to 
A'Oi  respectively.  Then 


144 

Hence 
cos  Oi  — 
and 


FOR    EXGlXKERS. 


K0t 


'°i  -  A'K 


0,  02. 


vers  Oi  =  1  -  cos  0,  =  - 
Transposing,  we  have 


i  v_  #   vers 


(20) 


and 


E  cos  I V  -  7/a  (cos  0,  -  cos  j  F) 
vers  01 

E  cos  t  F  -  g.  vers  Q, 
cos  Oj  —  cos  i  F 


Also,       .ffF  =  F03  sin  ^  F  =  (E  +  #„)  sin  |  F; 

A'H  =  KO.,  =  Q!*},  sin  0,  —  (/?i  -  #a)  sin  OL 
Hence     ^'  F=  T,  =  (^  +  7^2)  sin  |F+  (/A  -  tfa)  sin  0,.     (21) 

Since      cot  iOi  =    -  ,  we  have,  from  (20)  and  (21), 

vers  0, 


cot 


cos     F 


,Bn 

74  vers     F' 


In  the  figure  the  curve  is  flattened  at  the  ends,  but  either  or 
both  ends  may  be  sharpened,  the  equations  applying  without 
change. 

Supposing  the  vertex  distance  2i7to  be  known,  we  may  assume 
Itt  and  7?2  and  find  <9i  from  (20),  then  Tl  from  (21). 

Again,  we  may  assume  /£,  and  2\  ,  and  find  Oi  from  (22),  then 
II,  from  (20). 

Again,  we  may  assume  723  and  0H  and  find  7?j  from  (20),  then 
7T,  from  (21). 

Finally,  we  may  assume  2\  and  Oi,  and  find  7ija  from  (32),  tlu-n 
7?!  from  (20). 

After  finding  the  unknown  quantities,  we  may  lay  out  the 
curve  A'Piin,  consuming  out1  half  of  the  vertex  angle  F.  Then, 


COMPOUND    CURVES.  145 

letting  B'E  =  Rl  ,  P*C  -  7?2,  B'V=  T,  ,  and  B'EP*  =  0,  ,  we 
may  assume  values  without  reference  to  the  corresponding  values 
in  the  curve  A'P\in\  and  find  and  lay  out  a  curve  mP^B',  differ- 
ing to  any  possible  extent  from  the  curve  A' Pitn. 

Otherwise,  drawing  the  tangent  LmN  (Fig.  82),  we  have 

mL  V  -  mNV  =  k  V,     mL  =  T"  =  E  cot  \V \     L  V  =       E 


and  A'L  =  T  =  A'V  -  LV  -  T,  - 


sin  \V' 
E 


sin  \V 


With  the  vertex  angle  mLV  =  \V  and  the  tangents  A'L  and 
Lm  a  compound  curve  A'Ptm  of  two  branches  may  be  located  as 
usual,  using  eqs.  (1)  and  (2).  Similar!}'  for  the  curve  viB. 

It  is  evident  that  a  compound  curve  of  any  number  of  branches 
and  having  in  general  any  radii  may  connect  two  fixed  tangent 
points.  We  may,  for  example,  lay  out  all  but  two  of  the  proposed 
number  of  branches  ending,  we  will  suppose,  at  m,  Fig.  82, 
Then  run  out  and  measure  or  compute  mN\  likewise  NB' .  Then, 
observing  that  the  vertex  angle  at  .ZVis  equal  to  V  less  the  central 
angles  already  run  off,  we  may  connect  the  tangent  points  m  and 
B'  as  usual,  thus  completing  the  curve  A'mB'. 

Let  us  suppose  the  compound  curve  to  pass  through  the  middle 
point  of  the  simple  curve  AmB,  Fig.  82,  of  radius  AO  =  R,  con- 
necting the  tangents  AV  and  BV. 

Then  mV  =  E  =  7?(sec  \V  -  1), 

or  Ecoa  \V  —  R(l  —  cos  $V). 

Multiplying  by  -    i-  =  tan  $V,  we  obtain 

J£sin  %V=  2?(tan  \V  —  sin  £F). 

Now  since  these  equations  express  the  relations  existing  among" 
the  elements  R,  V,  and  E  of  a  simple  curve,  and  since,  too,  eqs. 
(20)  and  (21)  express  the  relations  among  the  elements  of  a  com- 
pound curve,  it  is  plain  that  if  we  substitute  in  eqs.  (20)  and  (21) 
the  value  of  E  (or  of  ^sin  £Fand  of  ^cos  |F)  drawn  from  these 
equations,  the  resulting  equations  will  express  the  relations  among 
the  elements  of  a  compound  curve  that  vi  ill  pass  through  the 
middle  point  m  of  a  simple  curve  of  radius  R  and  central  angle  V* 


146  FIELJKXTANU.'.L   FOR   ENGINEERS. 

We  thus  find  1 

(72  — 
versO,  =        - 


_  .     .     (33) 

vers  Ot 

7?vers£F-  ft  vers  0, 

or  ft,  =  -        —  TTF  —  —  •    ! 

vers  \V  —  vers  0j 

Also,   7',  =  7^  tan  \  F+  (ft  -  7*a)  sin  Ot  —  (72  -  723)  sin  \  V.     (24) 
But      T,  =  FA  +  ^.4'  =  R  tan  £F-f  ^^1'. 

Hence 

AA'  =  d  -  (Rl  -  723)  sin  0,  -  (R  -  7?a)  sin  i  F.   .     .     (25) 

These  results  are  of  course  easily  deduced  from  the  figure  in  the 
same  way  as  results  were  derived  from  Fig.  69  and  others. 
For  example, 


and 

r^^y-M^l'^tf  tan  |F+(ft—  ft,)  sin  0,-(ll-Rz,  sin  \V. 

Example.—  Let  0,  =  24°  15',  ft  =  905,  and  F=  81°. 

It  is  required  to  find  the  radius  PiO?  —  ft  of  the  second  branch 
of  the  compound  curve  passing  through  the  middle  point  of  a 
simple  curve  of  radius  H  =  630  (Fig.  82).  We  have 


V=K  (sec  -|F-1)  cos  £F=.K(l-cos  £F)  =  21  vers 

=  630  X  .23959  =  150.94; 
J2»  vers  0,  =  905  X  .08824  =    79.86. 
Also, 

cos  d  -  cos  -|F  =  .15136. 


COMPOUND   CURVES, 


147 


Then 

R9  =  71.08^.15136  =  469.6; 

2.  If  the  radius  7?a  is  com- 
mon to  both  central  bran- 
ches, we  have  a  carve  of 
three  branches  shown  in 
Fig.  83.  The  same  equa- 
tions, (20),  (21),  and  (2'2), 
apply. 

Example.  —  Let 
V  =  80°  16',     0,  =  12°  40', 


and   7?3  =860. 


Then 


-  Ri  =  151.1. 


From  the  first  of   group 
(20)  we  find 


FIG.  83. 


-  IBM  X  .9756 


,7o4o 

.-.     #=1129.8-860  =  269.8. 
(21)  gives 
T,  =  (E+  It,)  sin  -|F-f  (7?,  -  Rj  sin  0, 

=  1129.8  X  .6446  +  151.1  X  .2193  =  728.27  +  33.14  =  761.4. 
In  the  curve  MP^B  we  have 

E  =  269.8,     7V>3  =  R*  =  860. 
BCP?  =  d  =  17°  50'. 


Let 
(20)  gives 


269.8  X  .7645  -  860  x  .1874        45.10 
.04805" 


148 


FIELD-MANUAL    FOR    ENGINEERS, 


Also,  from  (21), 

J3V  =  2\  -  1129.8  X  .6446  -f-  151.1  X  .3062  =  728.27  -f  46.27 

=  774.54. 

3.   If  the  end  branches  have  a  common  radius,  R.,  and  central- 

angle,  Oi ,  and  the  central 
branches  a  common  radius, 
/?3,  we  have  a  symmetrical 
curve  of  three  branches  shown 
in  Fig.  84.  Equations  (20), 
B'  (21),  and  (22)  apply. 

I.  Suppose  the  curve  is  flat- 
tened at  the  ends. 
Example.  — Let 
V  -  67°,     Rl  =  3437.7, 


and 


:=  1432.4. 


0 


It  is  required  that  the  com- 
pound curve  shall  be  tangent 
at  the  middle  point  of  the 
simple  curve  whose  radius  is 

R  -  2291.8. 
We  have 


FIG.  84. 
E  =  R  exsec  \V  -  2291.8  X  0.1992  =  456.53. 

Eq.  (20)  gives 


cos  Ol  = 


Bl  - 


3437.7  -  1575.1 


_ 


—£_£,)  2005.3 

...     o»  =  21°  45'     and     Oa  =  V  -  201  =  23°  30'. 

Now,  from  (21), 

T,  =  (E  +  7?2)  sin  %V+(Ri  -  R*)  sin  0, 

=  1888.9  X  .55194  +  2005.3  X  .37056  =  1042.56  -f  743.08 

=  1785.64. 
The  tangent  of  the  simple  curve  is 


COMPOUND    CURVES. 


149 


Hence  tbe  tangent  point  of  the  compound  curve  is  back  of  tLe 
tangent  point  of  the  simple  curve  268.72  feet. 

Example  2.—  Let  V  =  60°,  R  =  2000,  R,  =  1200,  and  A  A1  =  d 
=  240.  To  find  E,  ri\  ,  0,  ,  and  St. 

W-  lessee  IV  -  2000  X  .1547  =  309.4.  .'.  E  +  R*  =  1509.4. 
Ti  =  rf  +  fltan  $  F  =  d  -f  2000  X  .57735  =  240  +  1154.7=  1394.7. 


"l  »°l  = 


1394.7  -  1509.4  X  .5000 


640 


1077T8 


_ 


.-.   01  =  19°. 

Now  (20)  gives 

267.94  -  1200  X  .0795       172.54 


.05448 


,05448 


_ 

-     OlOl.U. 


Observe  that  E  cos  £F=  267.94  is  computed  in  finding  cot 
If  Oi  is  given,  we  have,  from  (23), 


vers  0, 

Then  find  Tl  (and  E  if 
desired)  as  before. 

II.  The  curve  sharpened 
at  the  tangents. 

This  will  occur  when  the  »', 
compound  curve  is  a  moder- 
ately easy  connecting  curve 
between  two  tracks,  the 
curves  being  sharpened  at 
the  ends  to  fit  the  frogs. 

Example  1.— Let  V  —  27° 
40',  and  E  =  45.  Assume 
R}  =  744,  and  Ol  =  6°  34'. 
Eq.  (20)  gives 

-  43.695  -  4.881 

.022444 
=  1729,4. 


150  FIELD-MANUAL   FOH   ENGINEERS. 

Eq.  (21)  gives 
r,  =  1774.4  x  .2391  -  985.4  X  .11436  =  424.26  -  112.69  =  311.6. 

Example  2.— Let  V  =  63°,  and  E—  270.     Assume  ^  =  744,  and 
0,  =  20"  31'. 
Eq.  (20)  gives 

_  270  X  .85264  -  744  X  .06333  _  183.10  _ 
.08403  "  .0840^  ~ 

Eq.  (21)  gives 

AV  -  T,  =  2448.9  X  .5225  -f  1434.9  X  .35021 
=  1279.6  -f  502.5  =  1782.1. 

Problem. — To  draw  a  tangent  to  two  given  curves. 

Let  AP\  and  J3P3  be  the  curves,  AO  =  #,,  and  BO'  =  It*. 


FIG.  86. 

I.  When  the  curves  are  visible  from  each  other. 

In  this  case  a  point  Pi  may  be  found  by  trial  on  one  of  the 
curves,  through  which  if  a  tangent  line  be  drawn,  it  will  be  tan-: 
gent  to  the  other  curve  also. 

II.  "When  the  curves  are  not  visible  from  each  other. 

(A)  Suppose  the  distance  between  the  centers  00'  =  c,  and  the 
direction  of  00'  to  be  known.  Let  DOO'  —  0. 

Run  any  convenient  radius  AO.  Then  the  angle  AOO'  is 
known.  Let  Pil\  —  a  represent  the  tangent  line,  and  suppose 
O'D  to  be  drawn  parallel  to  it.  Then 

DO       K,  -  fa 


COMPOUND    CURVES.  151 

Also,  1\1\  =  DO1  =  c  siu  0.   , 

Then  A01\  =  AOO'  -  2)00', 

which  gives  the  length  of  the  arc  APi,  and  consequently  the  posi- 
tion of  Pi. 

P,Pi  makes  an  angle  with  the  tangent  at  A  equal  to  AOPt,  and 
therefore  PiP2  is  determined  in  every  respect.  If  more  conven- 
ient, run  any  line  AB,  and  measure  the  angle  OAB—  A.  Suppose 
BK  to  be  drawn  parallel  to  00'  ,  and  let  ABK  —  B.  We  have 


c 
Then  AOO'  =  AKB  =  180°  -  (A  +  B). 

Now  find  0,  AOPi,  and  P,P2  =  a  as  before. 
(B)  Suppose  that  00'  is  not  known. 

Run  and  measure  any  line  AB  =  d,  and  the  angle  BAK  =  A. 
Then,  in  the  triangle  ABK,  we  have 


AB  =  d,     AK  =  #1  -  #„  ,     and    JOS  =  4. 
Hence,  Chap.  Ill,  formula  (8), 

sin  A  sin  J. 


tan  B  = 


AB  d 

-j-=  -  cos  A  -  cos  A 

A  A  lii  —  -« a 


Then  i*- »T  =  ^^  =  («,  -  A)"4=4 

sm  5  sin  Z> 


Now  proceed  as  before. 

We  have  already  considered  generally  the  problem  of  connect- 
ing two  straight  lines  by  a  compound  curve  of  two  or  more 
branches. 

We  now  come  to  consider  the  problem  of  connecting  two 
curves  by  a  third  curve,  thus  forming  a  compound  curve  of  three 
branches. 

Problem. — To  locate  a  curve  tangent  internally  to  two  given 
curves,  as  shown  in  Fig.  87. 


152 


FIELD-MANUAL    FOR    ENGINEERS. 


Let  AOi  —  Ri,  and  (702  —  122,  represent  the  radii  of  the  curves. 

I.  Suppose  the  tangent  point  on  either  curve  to  be  given.  Let 
A  be  the  point. 

By  construction.  Make  An  —  (70s  =  It?.  Draw  nO-^  and  at  the 
middle  point  of  w02  erect  a  perpendicular  to  it  to  meet  AOi  in  0. 
0  is  the  center  and  A0=  JR  is  the  radius  of  the  connecting  curve. 

Calculation. — Run  and  measure  a  line  from  A  to  any  point  (7  on 
the  other  curve.  Let  AC  —  a. 


Measure  the  angles  PC  A  =  C,  and  CAO  —  A  (P  is  on  CO*  pro- 
longed). 

Draw  the  perpendiculars  CH  and  0*K  to  ^40i. 

Evidently  the  angle  between  (702  and  AOi  is  equal  to  ^1  —  6y. 

Therefore 

AK  =  AH  +  J7#  =  «  cos  J.  +  It*  ccs  (  J.  -  6Y). 
Similarly,  J?03  =  «  sin  A  +  J?3  sin  (A  —  C}. 

Now  Kn  =  AK  —  An  =  AK  -  It?.     Let  KnG*  -  71.     Then 


Kn 

; 
cos  11 


tan  n  •=  -jf—\     n 

Finally,  AO  =  DO  = 


Kn 


Kn 


cos  n      2  c< 
4- nO  =  II 


.,  . 

r-f-rc6s,2n 


153 

If  ACOi  can  be  made  a  straight  line, 

(7—0,  AK  —  (a  -f  74)  cos  A,     and    KO*  =  (a  -f  R^)  sin  A. 


We  Lave  7i002  =  180°  —  2n,  which  gives  the  central  angle,  and 
consequently  the  length  of  the  connecting  curve. 

To  find  OiOi  —  c  :  Find  KO*  and  AK  as  above.     Then 


d,01r=:IJ     and 


If  the  tangent  point  D  on  the  curve  of  shorter  radius  is  given, 
lay  off  on  Z>03  prolonged  a  distance  Dn  =AOi  =  Hi,  and  proceed 
precisely  as  above. 

II.  Suppose  the  radius  of  the  connecting  curve  to  be  given. 
Consider  any  radius  of  either  of  the  curves  that  will,  when  pro- 
longed, intersect  the  other  curve.  Let  O^C  represent  such  a 
radius,  and  O^CP  the  same  line  prolonged. 

Measure  CP  =  b,  and  OjP02  =  P.     Let  P0,0^  =  Olt 


sin  P                       sin  P 
tan  Oi  —  -7777 = — - 


(See  Chap.  Ill,  formula  (8).) 
Then 

P0*0l=m*  -(P+Oj.     Also,      Ok'&  »  •  « 
The  triangle  00,  0*  gives 


cos 


Then  AOi P  =  180°  -  (PO.O^  -f  00,0^. 

This  gives  the  distance  from  P  to  the  tangent  point  A 


154 


L    FOR    ENGINEERS. 


Also, 
tan  0,00.,  = 


sin  00!  0a 


sin  00,02 


-  cos  00,0, 


0,0a 


cos  00,  0a 


which  gives  the  length  of  the  required  curve. 

If  C  is  at  the  intersection  of  the  curves,  b  =  0;  otherwise  the 
same  formulas  apply. 

The  above  solution  applies  whatever  may  be  the  relative  posi- 
tion of  the  given  curves. 

When,  however,  the  given  curves  are  tangent  internally  there 
is  no  connecting  curve. 

Problem. — To  locate  a  given  curve    tangent  externally  to  two 
given  curves,  as  shown  in  Fig.  88. 

0 


Q, 


FIG  88. 
Let  A  be   the  given  tangent  point.     Let  A0t  =  Hi,  and  7?0-2 

Make  An  —  _Z?02,  and  at  the  middle  point  of  ?i0a  erect  a  per- 
pendicular to  w0a  to  meet  AOi,  prolonged  in  0.  0  is  the  center, 
and  AO  —  It  is  the  radius,  of  the  required  curve. 


COMPOUND    CURVES.  155 

ion.  —  Run  the  line  AC  =  a,  as  in  tlie  preceding  prob- 
lem, and  suppose  the  perpendiculars   (7/7  and  0*K  to  be  dra\vn. 
Let  AOL  =  C,  and  CA01  =  A.     Then 

AK  =  AH  +  HK  =  a  cos  A  +  R*  cos  (J.  -  C), 
and         KOi  =  a  sin  A  +  R*  sin  (^4  —  (7). 
Now        A"//  =  ^l?z  -  AK  =  #3  -  J7f. 

Let  jfiT»0a  =  w.     Then 

Kn 


tan  7i  = 


_ 

~ 


Kn  '  cos  ?i  ' 

A^  Kn 


cos  /<-  ~  2  cos2  7*  ~~  1  4-  CT>S  2n' 
Finally,       AO  =  »0  -  An  =  7J-0  -  #2. 

Suppose  tlie  radius  of  the  connecting  curve  AO  =  J?  to  be  given, 
and  the  position  of  the  tangent  points  and  the  length  of  the  curve 
required. 

Draw  COi  intersecting  the  curve  AD  in  P.  Measure  CP  —  />, 
and  the  angle  0;<702  =  (7.  Let  tfO.C^  =  OL  Now 

sin  C  sin  (7 

'  (    p'    '  eq-  ())' 


___COSC       -g-    -cosC 
Then  (70,0,  =  180°  -  «7  +  0.). 

Also,  0,0,  =  .  =  ft*^, 

The  triangle  OOiOt  gives 


Then  AOiP=  OOiO*  -  COiO9. 


156 


FIELD-MANUAL   FOR   ENGINEERS. 


(X 


This  gives  the  position  of  the  tangent  point  A  with  reference  to 
P. 

The  above  solution  applies  whatever  the  relative  position  of 
the  given  curves  may  be. 

When,  however,  the  given  curves  are  tangent  internally,  or  one 

of  them,  prolonged  at  pleas- 
ure, lies  entirely  within  the 
other,  there  is  no  connect- 
ing curve. 

Problem.  —  To  locate  a 
curve  tangent  to  two  given 
curves,  externally  to  one 
and  internally  to  the  other, 
thus  forming  a  compound 
and  a  reverse  curve. 

'LetAOl=Rl>  CO*  --=R*, 
A0=  E. 

Let  A  be  the  given  tan- 
gent point. 

By  Construction. — On  the  radius  0\A  prolonged  make  An  — 
COi  =  Ry.  Draw  nOi,  and  at  the  middle  point  of  nO?  erect  a 
perpendicular  mO  to  meet  AOi  in  0. 

0  is  the  center  and  AO  =  DO  =  R  is  the  radius  of  the  required 
curve. 

Calculation. — Run  the  line  AC  —  a,  as  in  the  preceding  prob- 
lems, and  suppose  the  perpendiculars  CH and  0*Kto  be  drawn. 

Let  ACL  =  C,     and     CAO,  =  A. 

Then         AK  =  AH -j-  HK  =  a  cos  A  -f  .#a  cos  (A  -  C), 
and.  KO*  —  a  sin  A  -f  #3  sin  (A  -  C). 

Now  Kn  =  An  -f  AK  =  R*  -f  AK.    Let  KnO*  -  n. 
Then 


KO*  Kn 

tan  n  =   -7^—;    nO*  —  — 
Jfrj,  cos  n 


.-.       Tgii,  \j%          Kn 

nO  —— =  — 

cos  n      2  cos2  n 


Kn 

1  +  cos  2n 


Finally, 


AO  -  nO  -"An  =  nO  - 


Suppose  the  radius  of  the  connecting  curve  AO  —  R  to  be 
given,  and  the  position  of  the  tangent  points  and  the  length  of 
the  connecting  curve  required.  Draw  CO,  intersecting  the  curvu 


COMPOUND  CURVES. 


157 


AD  in  P.     Measure   CP  =  b   and   the   angle    0,CO^  =  C.     Let 
COiO*  =  #,.     Now,  Chap.  Ill,  formula  (8), 


tan  0,  =  77- 


sin  0 


sin  C 


Xgi  _  cos  0       L^1J  _  cos  c 
Then  <70a0,  =  180°  -  (C  +  0t). 

Also,  0i  0a  =  C  =  J?2  - — 7T. 

sin  #i 

The  triangle  00i02  gives 

cos  00102  = 


or  cos  00!  03  =  {~~ 

Then  AO,P  =  180°  -  ((70,  02  +  00, 0a)« 

This  gives  the  position  of  the  tangent  point  ^L  with  reference 
to  P. 

The  above  solution  applies  for  all  positions  of  the  given  curves. 
When,  however,  the  curves  are  tangent  externally  there  is  no 
connecting  curve. 


Problem. — To  find  a  curve  tangent  to  a  given  curve,  and  to 
straight  line,  which  intersects  the  curve.     (Fig.  90.) 


158  FIELD-MANUAL   FOR   ENGINEERS. 

A.  Let  AO  =  11'  =  the  radius  of  the  given  curve  A  Vm,  and  let 
BD  be  the  line. 

The  radius  Hoi  the  required  curve  may  be  given,  or  the  tangent 
point  A  on  the  curve,  or  the  tangent  point  B  on  the  line. 

I.  Given  the  radius  AC  =  R.  Draw  VO.  Suppose  FAKL  and 
CH  drawn  parallel  to  BD.  Measure  the  angle  PVD  (which  is 
equal  to  DOV)  and  represent  it  by  A. 

Let  ACB  ~  AOD  —  0,  and  Dm  —  11'  vers  A  -  a. 

Now                                mL  —  BK  =  Dm. 
That  is,        11'  vers  0  —  It  vers  0  =  a,     or  vers  0  =  — . 

VB  =  VD  -  CH  =  11'  sin  A  -  (Rr  -  E)  sin  0. 
This  gives  the  position  of  B. 
Also,  AOV  =  AOD  -  VOD  =  0  -  A. 

This  gives  the  position  of  A.     From  A  or  from  B  the  curve  may 
be  run. 
We  have,  from  above, 

R'  sin  A  -  VB 


sin  0  = 


Whence  tan  £0  = 


R'  -  U 

vers  0  a 

sin  6  ~  Wsm  A  -  VB' 


II.  Suppose  the  tangent  point  A  to  be  assumed. 

Let  AOV  =  B.     Then  AOD  =  A  -f  B  =  0  is  known. 

We  have,  from  above, 

R  =  R'  - 


vers  0' 

BV  is  found  as  above. 

III.  Suppose  the  tangent  point  13  to  be  assumed.     Let  VB  =  b. 
Substituting  this  value  of  VB  in  the  value  of  tan  \0t  we  have 


.       . 
R'  sin  A 

Then  R  =  It' 


-. 
vers  0 


COMPOUND    CU FIVES.  5 

Xow  0  and  R  give  the  length  of  the  arc  AB  and  the  tangent 
point  A. 

Draw  Afim,  which  is  a  straight  line  as  shown  above. 


Now  DBm  =  AB  V  =  _ 

mD  —  R '  vers  A  =  «,     and     ED  —  YD  —  VB  =  R'sh\A-  VB. 

mD  a 

Plien         tan  mBD  =  -^^,     or     tan  \O  —  — — ; —    — ^^, 


as  already  shown. 

B.   Suppose    the   required    curve  to  be   tangent    to   the    given 
curve  externally,  as  also  shown  in  Fig.  90. 

Let  the  central  angle  AC'S'  =  180°  —  0,  and  AC'  =  r. 

Then  ACB  -  AOD  =  0. 

I.  Suppose  the  radius  AC'  —  r  given. 
We  have  mL  -  B'C'  -  C'F  =  Dm. 

That  is, 

R'  vers  0  —  r  —  r  cos  0  =  a,     or    R'  vers  0  —  2r  -\-r  vers  0  =  a. 
Hence 


This  gives  the  position  of  B  '. 

Also,  AOV=  AOD  -VOD  =  0  -  A. 

This  gives  the  position  of  A.     From  A  or  B'  the  curve  may  be 
run.     B'  is  also  given  by  the  angle  AC'B'  =  180°  —  0. 

We  have,  from  above,     sin  0  = 


R    -f  r 


Whence          tan  iO  =  ^ 

sin  0        R'  sin 


160 


FIELD-MANUAL   FOR   ENGINEERS. 


II.  Suppose  the  tangent  point  A  to  be  assumed. 
B.     Then 


A+B=0,     and     AC'B1  =  180°  -  (A  + 
is  known.     Then,  from  above, 

_  K'  vers  0  —  a 
1  -}-  cos  0     ' 

B'Vis  found  as  above. 

III.  Suppose  the  tangent  point  B'  to  be  assumed. 


Let 


Then 


VB'  =  V; 


r  = 


R'  vers  0  —  a 
1  -f  cos  0     ' 


=  180°  -0 


r  and  0. 

Give  the  length  of  the  arc  B  'A  and  the  tangent  point  A. 

C.  Suppose  the  required  curve  to  be  tangent  to  the  given  curve 


FIG.  91. 


internally,  the  centers  of  the  two  curves  being  on  opposite  sides 
of  the  line.     (Fig.  91.) 


COMPOUND   CURVES.  161 

Let  DOV  =  DVP  =  A,  and  Dm  =  72'  vers  J.  =  a,  as  before, 
Let  the  central  angle  ACS  —  180  —  0. 

Then  407)  =  0. 

1.  Let  the  radius  4(7  =  r  be  given.     Suppose  FAKL,  CH,  and 
C'U'  drawn  parallel  to  #7).     We  have 

mL-\-  BC  +  <7A'=  Dm. 
That  is, 

72'  vers  0  +  r~\-  r  cos  0  =  a,     or     72'  vers  0  -f  2;-  —  ?•  vers  0  =  a. 
Hence  vers  0  —  •=  -  . 

-ii     —  T 

Then  yiOF=  PTOZ>  - 

which  gives  the  position  of  the  tangent  point  A. 

Also,       F#  =  VD  -  CH=  R1  sin  A  -  (R'  -  r)  sin  0, 

which  gives  the  position  of  B,     From  A  or  from  B  the  curve  may 
be  run. 

R'sinA-VB 
From  above,     sin  0  —  --  W^r  -  ' 

vers  0  a  —  2r 

tan    °  =  -- 


II.  Suppose  the  tangent  point  A  is  assumed. 

Let  AOV  =  B.     Then  ^OD  =  4-#-  Ois  known. 

We  have,  from  the  above  equation, 


a  —  R'  vers 


1  +  cos  0     ' 

Then  FT?  as  given  above. 

III.  Suppose  the  tangent  point  B  to  be  assumed. 
Let  VB  =  b.     Eliminating  the  functions  of  0  from  the  equa- 
tions involving  sin  0  and  vers  0,  we  find 

_  A  _  (R>  sin  A  ~  b? 
T  ~  ~2  Ut'  -  2a      ' 


162  FIELD-MANUAL    FOtt   ENGINEERS, 

vers  0  a  —  2r 


Then  tan  \0  — 


sin  0       R'  sin  A  —  b' 


r  and  0  give  the  center  C  and  arc  BA  and  the  tangent  point  A. 
D.   Suppose  the  required  curve  to  be  tangent  to  the  given  curve 
externally,  as  also  shown  in  Fig.  91. 
Let  the  central  angle  AC' B'  =  0. 
I.  Suppose  the  radius  AC'  =  r  is  given. 

We  have  mL  +  B'F  =  Dm. 

That  is,  R'  vers  0  -\-  r  vers  0  =  a. 

Hence  vers  0  = 


R'  +  r 
Then  AO  V  =  VOD  -  AOD  =  A  -  0, 

which  gives  the  position  of  the  tangent  point  A. 
Also, 

VB'  &  C'H'  -  VD  =  (Rr  -f  -r)  sin  0  -  R'  sin  A, 

which  gives  the  position  of  Br. 

From  A  or  from  B'  the  curve  may  be  located. 

R'smA+  VB' 
From  above,  sin  0  =  -   — ^7 — -        — . 

R    -\-  T 

vers  0  a 

Then  tan  ±0  =  -—      = 


sin  0        It 'sin  A  +  VB'' 
II.  Suppose  the  tangent  point  A  is  assumed. 
Let  AO  V—  B.     Then  0  =  AC'B'  =  AOD  =  A—  B  is  known. 
Then  equation  above  gives 


-—. 

vers  0 

VB'  is  found  as  above. 

III.  Suppose  the  tangent  point  B'  is  assumed.     Let  VB'  —  b. 

We  have,  from  equation  above, 


tan  10  =  ^ry-r 
R'  sm 


COMPOUND   CUBVES. 


163 


Then  r  is  found  as  above. 

r  and  0  give  the  center  C",  the  arc  B'A,  and  the  tangent  point 


A. 


Also, 


70.1  =  VOD  -  AOD  =  A  -  0 


E     L 


D 


gives  the  arc  VA  and  the  tangent  point  A. 

Problem. — To  find  a  curve  tangent  to  a  given  curve  internally, 
and  to  a  straight  line,  that  does  not  in- 
tersect the  curve. 

A.   Let  AO  =  R'  —  radius  of  given 
curve  Am,  and  ED  =  given  line. 

I.  Suppose  the  radius,  A C=  BC=E, 
of  tlie  required  curve  AB  to  be  given. 

Draw  OD  perpendicular,  and  sup- 
pose AFKdr&vrii.  parallel  to  BD. 

Let  Dm  =  a, 


and  AOF=ACK  =  0. 

Now  BK=Dm  +  mF, 


K 


Fm.  92. 


or         R  vers  0  =  a  -(-  #'  vers  0,     or     vers  0  =   - 


It  -  R'' 

This  gives  the  length  of  the  arc  in  A  and  the  tangent  point  A, 
Also,  DB  =  FK  =  (#  -  R')  sin  0. 

This  determines  the  position  of  B  by  giving  its  distance  from  D., 
the  point  on  the  line  opposite  to  m. 
We  have,  from  above, 


sin  0  = 


BD 


R  -  R'' 


Whence 


vers  0         a 

tan  °  =         -  =      ' 


II.  If  the  tangent  point  A  is  given,  we  have  AOm  =•  0  given, 
and  from  the  equation  above  we  find 


vers  (? 


R'. 


164  FIELD-MANUAL   FOR   ENGINEERS. 

Also,  AK  =  R  sin  0, 

which  gives  the  position  of  B. 

III.   If  the  tangent  point  B  is  assumed,  let  BD  =  b,  and  we 
have,  from  above, 


We  also  have,  as  explained  above, 

Dm        a 


Also, 


vers  0* 


0  and  R  give  the  length  of  the  arc  BA  and  the  tangent  point  A. 

If  Dm  is  unknown,  run  and  measure  any  line  EP  =  b  and  draw 
PH.  Measure  EPL  (between  EP  and  OP  prolonged)  and 
PEL  =  E.  Then 

OLD  =  E  +  P,     and    POm  =  0  -  90°  -  (#  -f  P). 
Hence        DO  =  DH ~f-  HO  =  b  sin  E  +  #'  cos  0, 
and  Z>ra  =  J9#  —  inH  —  b  sin  E  —  R'  vers  wOP. 

B.  Let  the  curve  be  tangent  to  a  given  curve  externally,  and  to  a 

straight  line  that  does  not 
B      intersect  the  curve. 

Let  AO  =  R'  =the  ra- 
dius of  the  given  curve 
Am,  and  let  BD  be  the 
line. 

K  I.  Suppose  the  radius, 
AG-BG-r,  of  the  re- 
quired curve  to  be  given. 
Suppose GB  and  OD drawn 
perpendicular,  and  Cll 
and  FAK  parallel,  to  BD, 
The  notation  is  uniform  throughout  this  class  of  problems. 


COMPOUND    CURVES.  165 

Now  DO  ~  Dm  +  mO  =  a  +  Rr. 

Also,  DO  =  BC-+  110  =  r  +  (#'  +  r)  cos  0. 

Therefore          #  _|_  .R'  —  r  -f  (.R'  +  r)  cos  0, 

a  +  R'  —  r  2r  —  a 

or  cos  0  =  —  jj,         —  ,    and     vers  0  =  -jfTTT  • 

This  gives  the  length  of  the  arc  Am  and  the  tangent  point  A. 

Also,  BD  =  Oil  =  (R'  +  r)  sin  0. 

This  gives  the  position  of  the  tangent  point  B. 

2r  —  a 

Also,  tan  |0  =  vers  0  -r-  sin  0  —  —  =-=-  . 

x>// 

II.   If  the  tangent  point  ^4  is  assumed,  AOH  =  0  is  known,  and 
the  above  equation  gives 


_  -  cos  0} 

1  +  cos  0 

Then  #Z)  =  (R'  -f  r)  sin  0,     as  before. 

III.  Suppose  the  tangent  point  B  is  assumed.     Let  BD  =  &. 
We  have 

2r  -  a  b 

tan  \0  =  —  — 


2R'  -f  a' 

Observing  that  HO  —  DO  —  BC  =  a  -f  72'  —  r,  CH  =  b,  and 
CO  =  R'  +  7-,  we  deduce,  from  the  triangle  CHO,  or  from  the 
above, 


0  and  /•  give  the  length  of  the  arc  ^17?  and  ihe  tangent  point  A. 

"  WYE  "  PROBLEMS. 

The  preceding  equations  apply  at  once  to  "wye  "  problems. 
In  this  case  a  straight  line  and  two  curves  are  tangent  to  each, 
other,  or  three  curves  are  tangent  to  each  other, 


166 


1TELD-MANUAL    FOR   ENGINEERS. 


.—A  4°  curve,  radius  PjO,  =  It  =  1432.4,  ends  in  a 
tangent  PiP-j.  It  is  required  to  locate  a  curve  PPa  tangent  to 
tbe  straight  line  and  curve.  Draw  02/i  parallel  to  PiPa. 


Let 


FIG.  94. 
a  =  Ol     and     0,03Pa  =  0a. 


I.   Assume  the  radius  02P2  —  r  =  954.9  corresponding  to  a  6° 
curve. 


AT  w! 

Now       cos  Ol  =  -—  = 

01  02 


R-r 


477.5 


and 
Hence 


2387.3 

.-.   0x  =  78°. 46, 
09  =  180°  -  78°.46  =  101°.54. 


=  .2000. 


arc  fPl  =        -  =  1962  feet, 


and 


PP2  = 


arc        2 


feet. 


Also,     PiPa  =  w02  =  (R  +  r)  sin  0!  =  2339.1  feet. 

II.  Assume  the  tangent  point  P  on  the  curve  P,P. 
Then  Oi  is  given,  and,  from  equation  above,  we  have 

R(l  —  cos  0,) 


P,P2  =  (R+  r)sin  Ot. 


Also, 


This  determines  the  tangent  point  P2. 

III.  Assume  the  tangent  point  1\  on  the  line  PjPa. 


COMPOUND   CURVES.  167 

Let  Pi-Pa  =  c.     Then,  in  the  triangle  n0i0a, 


or  4Rr  =  c2,     or    r  =  —  . 

0i  and  02  and  the  position  of  Pi  are  now  found  as  in  case  I. 

Observing  that  the  radii  are  inversely  as  the  degree  of  the 
curve,  we  may  substitute  the  reciprocals  of  the  degrees  of  the 
curves  in  place  of  the  radii  in  the  formulas.  Thus 


cos  0,  = 


=  *  =  -2000-     ''•   0,=  78°.46. 


Example  £.— A  2°  curve,  radius  P10l  =  Rt  =  2864.8,  is  tangent 
at  P  to  a  3°  curve,  radius 
PO.,  =  It*  =  1909.9. 

It  is  required  to  locate      P,. 
a   0°    55'    curve,    radius 
P,0  —  R  =  6250.4,  tan- 
gent to  both  curves. 

In  the  triangle  OiOzO, 

0,0.  =  R,  +  R,, 

0:0  =  R    -  Rlf 

and    020  =  R   -  Ry. 

Hence  the  angles  may  be  found  by  the  general  formula 

fts  _U  c2  -  a9 


cos  a  = 


2bc 


Or,  observing  that  55'  =  \%  of  1°,  we  have,  as  explained  above, 

(12.   _   1\2  J_  /12.   _   1\2  _    fi   J_    r\V 

cos  0  =  V-i 


2(if  ~  i)(if  -  1) 
Or,  multiplying  through  by  2  X  3  X  11,  we  have 


.-.  0  =  75%2. 


168  FIELD-MANUAL   FOR  ENGINEERS. 

By  simply  transposing  this  equation,  we  have 

-0.  =  ™+^-™  =  .*,.^.,  s. 

Or  02  could  be  found  by  the  familiar  "sine  proportion."     Now 

Ol  =  180°  -  (0  +  02  )  =  61°. 52,     180°  -  02  =  136°.72, 
and  180°  -  0,  =  118°.48. 

Hence  P.P.,  =  7520    -*-  H  =  8204  feetJ 

PPi  =  11848  -f-  2    =  5924  feet; 
Pl\  =  13672  -H  3    =  4557  feet. 

II.   Suppose  PxOjP  is  assumed.     This  gives  the  position  of  PI. 
Then  00,  Oa  =  Oj  is  known.     Draw  09n,  making  Ptn  —  Pa0a. 
Let  0?iOa  =  C>02n  =  n.     Then 


sin  0i                            sin 
tan  n  =  —-^-  = 


Now  nOzOi  =  Oi  —  n,     00a7i  =  n. 

.'.   OOvP=2)i-0lt    and  P02P2=1800-OOaP=180°-(2>i-01). 
This  gives  the  length  of  the  arc  PP2  and  the  position  of  P2. 
OjOO,,  =  180  —  2-/z. 

This  gives  the  length  of  the  arc  PiP2. 
To  find  R,  we  have 

sin  0,  „  ,sin  Ol 

7i02  =  Oi02  — =  (H\  •+•  /la)—; 1 

Sill  /i  Sin  7i 


cos  ?i  sin  2n 

Finally,  P^O  =  ^  =  -K,  +  : 


COMPOUND   CURVES. 


169 


If    the    tangent    point    P2    is    assumed,   draw   0i/h,  making 
I\nl  =  P.  Oi ,  and  proceed  as  above. 

Example  3. — A  4°  curve,  radius  POi  =  R\,  is  tangent  externally 
to  a  5°  curve,  radius  P02  =  Ifa. 
It  is    required  to  locate   an    8° 
curve,    radius   PiO  =  R,   tan- 
gent to  both  curves. 

We  have 

cos  0_  (i 


_225+169-324_7_ 
2X15X13     ~39~ 

Therefore 


0  =  79°  40'  =  79°.  67. 

-  225  +  824  ~  169  -  —  - 

2  X  15  X  18       ~~  27  ~ 

.-.  0x=:450  17'  =  45°.28. 

Or  0!  may  be  found  by  the  "  sine  proportion."     Then 
02  =  180°  -  (0  -f-  0j)  =  55°  03'  =  55°. 05. 

II.  Suppose  0,  or  02  is  assumed;  0l5  for  example.     This  gives 
the  position  of  PI.     Draw  02n,  making  P,«,  =  P202. 
Let  0n02  —  002n  =  n.     Then 


tan  ;i  = 


COS  0t    — 


Then 

n0^01=n-0i,     and     0020!  =  02  =  nO^Ol  +  w  =  2n  -  0,. 
This  gives  the  arc  PPU  and  the  position  of  Pa. 
Also,  0  =  180°  -  (0,  +  0a), 

which  gives  the  length  of  the  arc  PjPa. 


170  FIELD-MANUAL   FOR   ENGINEERS. 

To  find  R,  we  have 


sm  n 


sin  n 


nO  = 


a)  sin  0, 


Then 


os  /i  2  sin  ?i  cos  ?i  sin  2n 

OPl  =  R  =  nO  -  nPt  =  nO  -  7?2. 


Problem.  —  Having  located  a  simple  curve  APB,  of  given 
radius  A0  =  E,  tangent  to  a  given  line  AG  at  A,  it  is  requi.ed 
to  run  a  compound  curve,  A'P'B',  of  two  branches  exterior  to 
the  simple  curve,  the  first  branch  having  the  shorter  radii  s  ai  d 
ending  in  a  given  parallel  tangent  A'O',  the  second  branch  b  'ing 
concentric  with  the  simple  curve  and  at  a  given  distance  from  it. 


Fw.  97. 


To  find  the  angle  of  the  first  branch.  Let  A'0l  =  J2i,  BE'  — 
PP'  =  D,  and  AE  =  d.  Let  AOP  =  A'01P  =  0.  Draw  0,K 
parallel  to  AG. 


COMPOUND   CURVES. 

Now     KO  +  R,  =  R  -f  d,     or    KO  =  R  -  R,  -f  d. 
Also,  0,0=  OP'  -  0,P'  =  R  +  D-Ri. 

KO         R  -  R,  +  d 
Hence  cos  O  —  -^-^r-  =  == -^ — ; — =r, 


171 


or 


vers  0  = 


D-d 


R  -  R,  -f  Z>* 


This  solves  the  problem  fully. 

The  formula  shows  that  d  must  be  less  than  D,  since  vers  0 
must  be  greater  than  zero;  and  that  the  less  is  the  difference 
between  d  and  D,  the  shorter  is  the  first  branch,  A'P',  of  the 
curve. 

When  d  =  D,  vers  0  =  zero;  therefore  0  =  zero,  the  branch 
A'P'  vanishes,  and  the  exterior  curve  becomes  a  simple  curve, 
concentric  with  the  curve  APB. 

Also,     EA'  =  KO,  =  00,  sin  0  =  (R  -  R,  -f-  D)  sin  0, 

which  gives  the  position  of  the  tangent  point  A',  and  the  distance 
apart,  measured  on  AG,  of  the  head-blocks  at  A  and  A'. 

When  d  =  zero,  AG  and  A'G'  coincide,  and  the  two  curves 
are  tangent  to  the  same  A  A'  G 

straight  line,  at  A  and  at 
A',  as  shown  in  Fig.  98. 

The  formula  for  this  case 
becomes 

Ters0 =___. 


B 


B' 
The  formula  shows  that 

0  increases  and  decreases 
as  R,  increases  and  de- 
creases. 

Hence,  in  order  that  the 
parallel  branch  B'P'  may 
be  comparatively  long,  and 
therefore  the  branch  A'P' 
correspondingly  short,  the  radius  7?,  must  be  correspondingly 
short,  and  the  frog  at  A'  sharp. 


FIG.  98. 


172  FIELD-MANUAL  FOR  ENGINEERS. 

Example. —  Suppose  a  number  9  frog  placed  at  A,  and  a  num- 
ber 7  frog  at  A'.  Let  BE'  =  D  =  22  feet. 

Required  the  angle  AOP  =  0,  and  the  distance  A  A'  between 
the  frogs.  In  this  case 

AO  =  R  =  744.0,     and     A'O,  =  Bl  =  439.4. 
.-.  R  -  It,  +  D  =  326.6, 

22 

whence          vers  0  =  — — .=  .06736,    and    0  —  21°  09'  =  21°.  15. 
o26.o 

AA'  =  (R  -  R,  +  D)  sin  0  =  326.6  X  .36081  =  117.84. 
The  degree  of  the  curve  A'P'  is  13°.  04. 

=  162  feet. 


If  AA  is  given,  we  have,  from  the  preceding  equations, 
sin  0  — r> — r— 77,     and     vers  0  —  ~ -— — 


-  vers  0  _    -P 
=    sin  0  ~  AA'' 

If  any  two  of  the  quantities  0,  D,  or  AA'  are  known  or 
assumed,  we  may  find  the  other  from  this  equation. 

We  observe  that  the  head-block  A'  must  be  placed  where  the 
main  track,  AG,  and  the  first  turnout  curve,  AP  are  sufficiently 
fur  apart  to  give  room  for  it.  This  limits  the  distance  AA' . 

Problem. — Having  located  a  simple  curve,  A'PB',  of  given 
radius,  A'0\  =  Ri,  tangent  to  a  given  line,  A'G' ',  at  A' ,  it  is  required 
to  run  a  compound  curve,  APB,  of  two  branches  exterior  to  the 
simple  curve,  the  first  branch  having  the  longer  radius  and  ending 
in  a  given  parallel  tangent,  AO;  the  second  branch  being  concen- 
tric with  the  simple  curve  and  at  a  given  distance  from  it. 

Let  AO  =  R,  A'Oi  =  R, , 

etc.,  as  before.     We  have 

R  —  KO  —  Ri  -4-  d,     or    KO  —  R  —  Ri  —  d , 


COMPOUND   CURVES. 


173 


and  00i  —  OP  -  OiP  =  R  -  Ri  -  D. 

Hence 

KO       R-  R,-d  d-  D 

cos  0  -=-^r  =  ^ ^  .  vers  0  — 


00!  ~  R  -  j«i  - 


FIG.  99. 

Since  vers  0  must  be  greater  than  zero,  d  must  be  greater  than 
D. 

In  case  D  —  zero,  the  arcs  BP  and  J9'P'  coincide,  and  the 
problem  is  the  same  as  that  given  under  Figs.  69  and  70,  and  we 
have 


as  before  shown. 

In  case  the  simple  curve,  in  the  two  preceding  problems,  is 
exterior  to  the  compound  curve,  the  same  solution  applies. 

These  problems  are  useful  about  yards  and  switches  when  it  is 
desirable  or  necessary  to  have  turnouts  from  different  switches 
merge  into  parallel  curves. 

The  radii  R  and  Hi  are  determined  by  the  numbers  of  the  frogs 
used  at  A  and  at  A'. 


174 


FIELD-MANUAL   FOR   EXGIXKEftS. 


To  locate  the  second  branch  of  a  compound  or  reversed  curve 
from    a    station    A    on    the   first 
^*r  branch  of  the  curve. 

T  P  T'       AP  and  PB  are  the  two  bran- 

ches of  two  curves  of  DI  and  7>2 
>B  degrees;   TPT'  the  common  tan- 
gent, and  A  V  a  tangent  at  A. 

Let  HI  and  w2  be  the  number  of 
chains    in    AP    and    PB    respec- 
tively.    Let    li    and   /a    represent 
the  long  chords  AP  and  PB. 
On  Then  the  angle 

APT=$nlDl, 

and  the  angle 

BPT'  =  $n,I),. 
FIG.  100. 


Hence,  by  formula  (8),  Chap.  Ill, 

sin  APB  sin 

tan  PAB=  - 


-f  n 


BP-^APB      £  +  C°Si 
BAP  +  ABP  =  180°  -  APB  = 


.     (26) 


Hence,  if  HiDi  +  W2Z)2  is  small,  we  have,  from  the  above  the 
approximate  formula, 


PAB  = 


nlDl 


For  reversed  curves  7)2  is  negative  and  the  formulas  become 

sin  Un-iDi  • 
tan  PAB  —  ; 


and 


.     .     (28) 


COMPOUND   CURVES. 

The  above  formula  is  easily  deduced  as  follows: 
BAP  +  A BP  =  180°  -  APE  =  APT  +  BPTf  = 


175 


In  the  triangle  ABP  the  angles  are  proportional  to  the  opposite 
sides  nearly.     Hence 


BAP  = 


BP 


InJJ^ 


Hi  4- 


Having  run  a  curve  with  the  center  0  and  radius  R  through  a 
given  arc  AB,   it  is  required  to   find   the      y\ 
radius  of  a  curve  with   center  0'  on  AO 
that  will  end  at  C  on  B'J  at  a  given  distance 
A  from  B. 

Solution.—  In  the  triangle  ABC  we  have 


BC=  a,     AB  =  c  =  2tt  sin  \0, 


and 
Then 


tan  BA  C  = 


sin  B 


cos  B 

a 


FIG.  101. 

Now     0'  +  WAV  =  180°,     and     0  +  2£.40'  =  180°. 
Equating  and  transposing  gives 
0'  -  0  =  2J5^0'  -  2(L!0'  =  2BA  C  -  2A,     or     0'  =  0  +  2A. 

Hence  AO'  =  <70'  *  (70^  =  (fi  -  «)(8—  A 

sin  Oi  \sm  Oi/ 


Example.—  Let  #  =  640, 
J5  ^  70°. 


-  a  =  100,      0  =  40°.      Then 


Now  c  =  1280  X  -3420  =  437.8;      -  =  4.378;  cos  B  --  .3420, 


176 


FIELD-MANUAL   FOR   ENGINEERS. 


Then 


.-.     ^  =  13°  06', 


and 

Finally,          CO  = 


0'  =  0  4-  %A  =  66°  12'. 
540  X  .6428 


.9205 


=  377.1. 


Problem. — Given  a  compound  curve  of  any  number  of  branches, 
to  find  the  length  of  the  tangents. 

A'  A  Vi  Vaxi  y3a?2  x3  Let  APi  P2 P3  (Fig.  102)  represent  a 
compound  curve  of  three  branches, 
the  central  angles  being  Oi,  0<2, 
and  03,  and  the  radii  7^,  7^2,  and 
J?3.  Draw  through  the  centers 
p3  parallels,  and  through  the  tangent 
points  perpendiculars,  to  the  tan , 
gent  at  A. 

02P2c  =  P2  F2z3  =  0i+  02, 
and 


02/1 


d     6 


FIG.  102. 

03P3e  =  P3V3x3  =  0!+  02  4-  03. 
Let  PiXi  =  h,     P*x*  =  U,     and     P3x,  —  t». 

We  have  tt  —  R>  vers  Oi  ; 

t,  -  ti  +  PJ  -  Pac  =  <,  +  7?a[cos  0,  -  cos  (0,  +  02)] 
=  Bi.  vers  0,  +  J?2[vers  (0,  -f-  0,)  -  vers  0,], 

or         U  =  (J?i  -  J2»)  vers  0,  +  7^  vers  (0,  +  02); 

«,  =  t*  +  P3(Z  -  P3e  =  1?,  vers  Oi  +  R2[vers  (0x  -\-  02)  -  vers  0i] 
4-  12,[vers  (0i  +  02  -f-  03)  -  vers  (0,  4-  02)], 

or         tt  =  (Ri  -  R*}  vers  0j  -{-  (J?2  -  J£3)  vers  (0,  -f  02) 
4-  1?3  vers  (0,  4-  02  4-  Ot). 

The  law  connecting  the  terms  of  these  expressions  is  very 
simple,  and  therefore  the  expression  for  any  tangent  may  be 
written  at  once. 


Now 

as  before  found. 


P,  V,  =  = 

sm  0, 


=  Bl  tan 


COMPOUND    CURVES.  177 


_          #2          _#,  ver80i-J-7Za[vers(0|+0a)-vers0i]  t 

2    2~ 


sin(0!-|-02)  yin  ( 

72  1  vers  0,+^2[vers  (0,  +  0a)  -  vers  0,]  -    («) 

__  +#3[vers(0.4-02  +  03)-vers(01  +  02)]     I 
sin  (0!  -f  0a  +  Oa) 

The  tangents  P!  V\  and  P3  F3  are  scarcely  needed. 
Again: 


Considering  any  pair  of  tangents,  AVs  and  PaF3  for  example, 
ami  noting  that  their  relations  to  the  radii  and  central  angles  are 
identical,  excep.t  that  they  stand  at  the  opposite  ends  of  the  curve, 
it  is  plain  that  the  formula  for  AV*  may  be  derived  from  that  of 
P2  Vy  or  w'ctf  versa,  by  simply  interchanging  7?i  and  7£a,  as  well  as 
Oi  and  02. 

For  the  same  reason,  the  formula  for  A  V*  may  be  derived  from 
that  of  P3V3,  or  vice  versa,  by  substituting  K3,  J?2,  and  R},  in 
order,  in  place  of  Hi,  7?2,  and  Jtf3>  in  order,  and  doing  the  same 
with  the  central  angles. 

This  amounts  to  interchanging  7?j  and  M3  as  well  as  0j  and  03. 

Thus  we  have 

_R,  vers  0,  -f  ^.[vers  (0,  +  09)  -  vers  Qa] 
sin  (.0»  +  0a) 

.Kg  vers  03  +  Jf?a[vers  (02  +  03)  —  vers  03] 
_  +ft[vers(0i  +  02  +  0  3)-vers(02+03 

sin(01  +  0a  +  03) 

Example.—  "Let  A01P1  =  Oi  —  38°,    PjOaPa  =  02  =  18°,    A0l 
=  fa  =  600,  and  P202  =  It,  =  900. 
Tofind^F2  andP2F2. 
From  the  above  formula  we  have 

600  X  .2120    -f  900  X  .2288        333.1 

~~  =~=      °L8> 


_  900  X  .048~94  -f  600  X  .39187  _  279.17  _       ,  „ 
.82904  ~  .82904  ~ 

If  we  draw  lines  parallel  and  perpendicular  to  I\V3  instead  of 
to  AV3t  as  in  Fig.  102,  \ve  may  compute  AVy,  AV3,  etc.,  directly. 


178 


FIELD-MAKUAL   FOK   ENGINEERS. 


Problem. — To  determine  a  simple  curve  tangent  to  a  compound 
curve  at  any  point. 

Let  P2  (Fig.  102)  be  the  point.     Make  A  A  —  P2F2  —  ^LF2. 
We  readily  find 

P  V   -  AV  -  (R*  ~  Iil^VerS  (Ol  +  Og)  ~(vers  Qi  +  vers  Q«)] 

sin  (0,  +  02) 

This  gives  the  tangent  point  A'.     Then 

A'E  =  R--A'V*  cot  ^(0,  -f  02). 

Observe  that  P2  represents  any  point  on  the  arc  PiP2. 

In  the  same  manner  the  elements  of  a  simple   curve  may  be 
found  tangent  to  a  compound  curve  at  any  point. 

Problem. — Two  curved  railway  tracks  cross  each  other.    To  find 

the  angles  at  the  crossings  of 
the  rails,  and  the  relation-; 
between  them.  Let  0  be  the 
center  of  the  track  BA  and 
DC,  and  0X  the  center  of  the 
track  AC  and  BD.  Draw  the 
radii  OA,  OB,  etc.,  OiA,  OJ1, 
etc.  OD  and  O^D  not  drawn 
on  the  figure. 

Let   AO  --=  R,  A01  =  R,, 
BO,  =  R,  +  g,  CO  .-  R  -f  g, 
and  OOi  =  c.    g  is  the  gauge 
of  the  tracks. 
The  triangle  ^100,  gives 


cos  A  — 


+  RJ  -  c\ 


(c  +  R- 


vers  A  — 

Similarly,  from  the  triangles  7?00i,  <700i,  and  7900,,  we  have 

7^2  +  (7?,  -f  fir)2  -  c,* 
cos  B  = 


vers  J3  = 


_  (c  -f  R  -  R,  -  f/Yc  -  H  -f-  A',  -4-  g) . 


COMPOUND   CURVES.  179 

72,2  +  (R  +  ffY  -  <? 
~3^BT7T" 

(c.  +  R,  -  R  -  g}(c  -  Ri  +  R  +  ff) 


(ft-f  f/)2  +  (72, 

and      cosl)= 


n  -  L 


+  g) 


These  equations  give  the  angles  sought. 

Any  two  of  these  equations  will  give  the  relation  between  the 
two  angles  involved. 

Thus,  equating  the  values  of  c3  in  the  first  and  third  equations 
above,  we  have 

R*-  +  (7^  +  </)2  -  2/2(7^  -f  g)  cos  B  =  R*  +  RS  -  2RK,  cos  A. 
From  this  we  readily  find 


Or,  adding   ^  J;  -  -  to  the  second  member  of  the  equation,  we 
M(Hi  -f-  g) 

have 

cos  B  =  ^  —  J—  cos  A  +  ^,  u«ry  nearly.      .      (30) 
-cvi  -f-  ^  Ji 

Similarly  for  any  two  of  the  angles. 

Writing  the  approximate  values  only,  which  are  very  nearly 
correct,  we  have 

cos  C  =  ^—.  —  cos  A  -f-  -—  .  very  nearly;  ...  .     (31) 

g  Mi 

+        '  verynearly;  (32) 


:'  '  (33) 


cos  7)  =  7—  -    cos  7?  +  —  -  —  ,  very  nearly;  .....     (34) 
-li  -f-  g  ii\  *f-  P' 


180 


FIELD-MANUAL    FOR    ENGINEERS. 


cos  D  =  H—  n  —  cos 
Mi  -\-  g 


A  +  g 


>  very  nearly. 


(35) 


Let  d  —  the  degree  of  the  curve  of  the  rail  AB,  D  that  of  CD, 
dr.  that  of  AC,  and  Dl  that  of  BD. 

In  case  the  angles  OAOi,  OBOlt  exceed  90°.     Let  A,  B,  etc., 


FIG.  104. 

represent  their  supplements,  which  are  the  angles  of  intersection 
of  the  rails.     Then 


vers  A  = 


cos  B  = 


Ri  +c}(R+  R,  -c) 


-  H*  -(B,  +.9)* 


and 


cos  C  — 


cos  J)  — 


(R 


+  g) 
g)*  -  (B, 


Since  these  values  are  tlie  same  as  in  the  former  case,  with  a 
negative  sigu,  it  is  evident  that  if  we  eliuiiuate  c  froin  them  we 


COMPOUND    CURVES.  181 

will  obtain  equations  the  same  as  equations  (30)  to  (35),  with  the 
sign  of  the  last  or  absolute  term  changed.     Thus  we  have 

cos  B  =  -^  —  -  —  cos  A  —  -—  ,  very  nearly;      ......     (36) 

-tii  +  g  & 

cos  C  =  ^  -  cos  A  —  ~,  very  nearly;  .......     (37) 

R  +  g  Mi 

ras  D  =  cos  A  ~     ~      '  very  nearly  ;  (38) 


cosD  =  -      -  cos  B  —  —  —  ;  —  ,  very  nearly;    .....     (40) 
R  -f-  g  A!  -f-  g 

l       cos  6'  —  -^  --  ,  very  nearly  ......     (41) 


R,  -f-  ^ 

We  may  find    results  practically  true  by  the  following  short 
and  simple  method. 

Let  L  in  Fig.  103  mark  the  intersection  of  AD  with  the  radius 
CO.    Now  the  sum  of  the  angles  in  the  triangles  A00t 
being  equal,  we  have 


B+J30A  +  AGO*  +  00,B  =  A  +  A 

.-.  B  =  A  +  AO^B  -  AOB. 
Now  BH  =  g,     and     BAH  =  A,  nearly. 

Let  JBTbe  the  intersection  of  the  arc  CA  and  the  radius  B0\. 

Then     All  =  -$—„  nearly,    and    AB  =  -r^-r,  nearly. 
tan  A  sin  A 


._„      AB  X  d  dg 

and  the  angle       AOB  =  ----- 


100  100  sin 


Hence  5  =  A  +  ----    -  -  .  .     (42) 

4        sin 


18x5  FIELD-MANUAL   FOE  ENGINEERS. 

Similarly,  C  =  A +/--(-?—-  -^l\  .     .     .     (43) 

100\tan  A        sm  AJ 

Example.— Let  R  =  1228,   ft,  =  764.     .-.  d  =  4°  40',  and  di  — 
T  30'.     Also,  g  =  4.708,  and  A  =  20°,  to  find  B  and  C. 
Eq.  (30)  gives 

cos  B=  .993875  X  .93969  +  .003834  =  .937869. 

.-.  £=20°  18'-|-. 
Or  eq.  (42)  gives 

B  =  20  +  .04708(20.606  -  13.644)  =  20°  +  0°.32777  =  20°  19'  -f . 
Eq.  (31)  gives 

cos  0=  .99618  X  .93969  +  .006162  =  .94226 

.'.  C  =  19°  34'. 
Or  eq.  (43)  gives 

cos  0  =  20  +  .04708(12.82  —  21.93)  =  19°. 57  =  19°  34'. 


CHAPTER  VII. 

REVERSED   CURVES. 

REVERSED  curves  are  mostly  used  in  connection  with  turnouts 
where  the  velocity  is  slow  and  space  an  object.  They  are  espe- 
cially objectionable  on  the  main  line  since,  adjacent  to  the  revers- 
ing point,  the  outer  rails  of  the  two  branches  are  on-opposite  sides 
of  the  track.  This  calls  for  the  elevation  of  both  sides  at  once, 
which  is  impossible,  and  so  the  track  must  be  level  at  that  point 
and  cannot  therefore  be  properly  elevated  at  any  point  near  it. 

Theorem. — The  reversing  point   of  a   reversed  curve  between 


FIG.  105. 

parallel  tangents  is  in  the  line  joining  the  tangent  points.  Let 
AMB  be  such  a  curve  reversing  at  M,  the  radii  of  the  two 
branches  being  equal  or  unequal. 

Draw  the  chords  AM  and  BM\  and  C7/ parallel  to  AD,  to  meet 
OB  prolonged  ;  also  the  radii  as  shown.  The  radii  CA  and  BO, 
being  perpendicular  to  the  parallel  tangents,  are  parallel,  and  the 

183 


184  FIELD-MANUAL   FOE   ENGINEERS, 

radii  CM  and  MO,  being  perpendicular  to  the  common  tangent  at 
Jfare  in  the  same  straight  line.     Hence 

ACM  =  BOM. 

.'.     90°  -  %ACM  =  90°  -  IBOM, 
or  A  MG  =  BMO. 

Hence  AM  and  MB  are  in  the  same  straight  line,  that  is,  M  is 
on  the  line  AB. 

I.  To  connect  two  parallel  tangents  by  a  reversed  curve  having 
equal  radii. 

Given  the  perpendicular  distance  between  two  parallel  tan- 
gents, BD  =  b  (Fig.  105),  and  the  common  radius  CM  =  MO,  to 
find  AD  =  a  and  the  chords  AM  and  BM. 

Let  AB=c.  Since  the  radii  are  equal  and  the  angle  A  C'M=BOM, 
the  triangles  ACM  and  BMO  are  equal,  and  AM  =  BM  =  \c. 

Hence  when  the  radii  are  equal  the  chords  AM  and  BM  are 
equal.  Draw  CE  perpendicular  to  AM. 

Then  AE  =  ±A  M  =  ±c, 

angle  BAD  =  ±ACM  =  ACE, 

and  hence  the  right  triangles  BAD  and  ACE  are  similar  and  give 
AC-t-AE=  AB-^-BD, 

or  R  -+-  ic  =  c  -T-  6. 


or    J2==,     or  6  =        .      .     .     (1) 
But  c2  =  a2  +  Z>«  ........     (2) 


-b),  or  b=2Jt  -  l/4IF-a>.    (3) 


When  any  two  of  the  quantities  «,  Z>,  c,  and  7v*  are  given  the 
other  two  are  readily  found  by  these  equations. 


REVERSED   CURVES.  1B5 

Again,  OH  =  OB  +  DH  -  ED  =  2E  -  b, 

and  CO  =  2R. 

Let  ACM  =  BOM  =  A. 

Then  cos  A  =  —  —  —  ,     or     vers  A  =  ^  .....     (4) 


AD  =  CH=2R  sin  .4,     AM  =  2.4#  =  2#  sin  £A,  ) 

/•  •     (5) 
and  ^15  =  2.4  3f  =  472  sin  |^L  ) 

Example.—  Let  b  =  20,-   and    #  =  741.7.         .'.     JD  =  7°.725. 
To  find  ^D,  JLJIf,  etc. 


vers  .4  =  -          =  .01348.         .'.     ^1  =  9°  25'  =  9°.417. 

1483.4 


Then  AD  =*2^  sin  ^4.    =  1483.4  X  .163613  =  242.70; 

AM-  2R  sin  %A  =  1483.4  X  .08209    =  121.77. 

9.417 
Also,      arc  AM  =  100,^^^  -  121.90. 

L  i/f/Q 

If  AD  and  BD  are  given,  then 

BD  BD  AB  AB 

tan  I  A  =  -p^;     AB  —  -  —  —  -\     and     R  = 

' 


sin  ^A       4  sin 
Example.—  Let  BD  =  13,  and  AD  =  200.     Then 

1  ° 

tan^=355  =  -065- 

.-.  ^1  =  3°  43'  08";     sin  %A  =  .06486; 
AB  =  =  20°'43; 


186 


FIELD-MANUAL   FOR   ENGINEERS. 


II.  To  connect  two  parallel  tangents  by  a  reversed  curve  hav- 
ing unequal  radii.     (Fig.  106.) 
We  Lave 


/I 


CO  =  CM  +  MO  =  R+r, 
and 
HO  =  OB+AC-BD=lt+r-b. 

r-b 


.'.  cos  A  —  - 


It  ±  r 


vers  A  =  ^r—  —  . 
R  -f  r 


FIG.  106. 


Then 

AD  =  CH  =  (R  +  r)  sin  A; 
AM  =  2R  sin  \A ;    BM  =  2r  sin  \A ; 

AS  =  AM+  BM  =  2(R  -f  r}  sin  ^A. 

Again,  let   AM  =  c',  BM  =  c",  ^5  =  c'  +  c"  =  c.     Then,  as 
above,  from  triangles  J.  (7,57  and  ABD, 


and,  from  triangles  BFO  and 


Adding  gives 


a2  + 


This  shows  that  for  the  same  distances  AD  and  BD  the 
o/"  the  radii  is  constant. 

1.  Given  the  perpendicular  distance  BD  between  tangents,  the 
radii  CM  -  R  and  MO  -  r,  to  find  AD,  the  chords  AM  and  J/5, 
and  the  central  angles  A. 


REVERSED    CURVES.  IS? 

Example.—  Let  BD  =  28,  It  =  955,  r  =  574. 

.-.  vers  A  =  Tf|T  =  .01831.     .-.  4  =  10°  59'. 
sin  J[  =  .19052,    and     sin  ^  =  .09570. 
AD  =  1529  X  .19052  =  291.50; 
AM  =  1910  X  .09570  =  182.79; 
BM  =  1148  X  .09570  =  109.86. 

2.  Let  AD,  BD,  and  R  be  given,  to  find  the  other  radius,  r,  the 
chords,  and  the  central  angles. 

Example.—  Let  AD  =  400,  BD  =  56,  and  R  =  900.     We  have 

tan  \A  =  ^  =  .14.     .-.  \A  -  7°  58'. 
AB  =  BD  -f-  sin  iJ.  =  56  -^  .13802  =  404.04; 
AM-2R  sin  $4  =  1800   X  .13860  =  249.48; 
and  BM  =  AB  —  AM  =  154.56. 

r  =    EM-^  sin      4  =  77.28  -s-  .13860  =  557.6. 


III.  To  unite  two  tangents  not  parallel  by  a  reversed  curve  of 
common  radius.  (Fig.107.) 

Given  the  angles  of  in- 
tersection at  D  and  E,  and 
the  length  of  the  common 
tangent  DE,  to  find  the 
common  radius  CM=  MO 
=  R  to  unite  the  tangents 
AD  and  BE. 

"We  have 


DM  —  R  tan  fa 

D;                   C 

EM  =  R  tan  fa 

FIG.  107. 

•'• 

DE  =  R(t&n  fa 

D  -f  tan  %E), 
)E 

tan  %D  +  tan  %& 
Example.—  Let  DE  =  250,  D  =  AOM=  24°,  ^=  BOM=  16°. 

tan  12°  -f  tan  8°  =  .35310. 
,-.  R  =  250  -*  .35310  =  708.02. 


FIELD-MAKUAL   FOR   ENGINEERS. 


IV.  To  connect  two  given  points  on  two  diverging  tangents  by 

a  reversed  curve  of   com- 
mon radius.     (Fig.  108.) 

Given  the  angles  BAD 
=  A,  ABE  =  B,  also  AB 
a,  to  find  the  radius  R  of 
the  curve  A  MB. 

Draw  CL  and  ONH  per- 
pendicular, and  CII  paral- 
lel, to  AB. 

Let 
HCO=BKO=CKA=K. 

Then 
HO  =  CO  sin  K=  2B  sin  K. 


H 


FIG.  108. 

Also,  HO  =  CL  +  NO  =  B  cos  A  +  R  cos  B. 

.-.      2^  sin  K=R  cos  .1+72  cos  J5, 
cos  J.  4-  cos  B 


T^ 
:  sin  K— 

4 

Again,    AL  =  R  sin  A  ; 
En  =  R  sin  5. 


=  Off  =  CO  cos  K=2RcosK', 


Adding  gives 


a  =  R(sm  A  +  sin  B  +  2  cos  IT), 


sin  A  -f-  sin  B  -f-  2  cos  1C 


ACK  =  ACL  +  Z(7#  =  J.  +  90°  - 
£  +  90°  -  .fiT. 


or 

Also, 
and 

The  foregoing  solution  requiring  barely  a  single  division,  ex- 
cept by  the  factor  2,  may  be  compared  with  Henck  Problem  40. 
Example.— Let  a  =  1500,  A  =  18°,  B  =  6°,  to  find  R. 

cos  .4  =  .95106 

cos  B  =  .99452         sin  A  =  .30902 

2)1.94558         sin  B  =  .10453 
sin  K  =  .97279    .'.  2  cos  K  =  .46340 

Then  R  =  1500  -f-  .87695  =  1710.47. 


REVERSED  CURVES. 


189 


V.  To  connect  two  diverging  tangents  by  a  reversed  curve, 
starting  at  a  given  point. 


FIG.  109 

(a]  Advancing  toward  the  intersection  of  tangents.     (Fig.  109.) 
Given  the  angle  of  intersection  AKB  =  CAL  =  K,  the  initial 
point  A,  the  distance  AK  =  «,  and  the  radii  CP  =  R  and  OP  =  r, 
to  find  the  central  angles  C and  0  of  the  reversed  curve  APB. 
First  Solution.  — We  have 


and 
Then 


NO 

cos°=co  =: 


BN  =  AL  —  AH  =  R  cos  K  —  a  sin  K. 
BN+r 


and     C  =  DCP  -  ACD  -  0  ~  K. 


Now  having  the  central  angles  C  and  0,  run  the  branch  AP 
subtending  the  angle  G ;  then  reverse  and  run  the  branch  PB 
subtending  the  angle  0. 

Example.'— Let  K-  24°  30',  AK=  854,  R  -  1440,  and  r  =  1146 

Then      AL  =  1440  X      .91  =  1310.4 
AH=    854  X  .4147  =    354.1 
BN=    956.3 
r  -  1146 


(Bn  +  r  =  2102.3)  -f-  2586  =  .81291. 
.*.     0  =  35°  37',     and     C  =  0  —  K  =  11°  07'. 

If  convenient  to   measure  AH  instead   of  AK,  the   solution  is 
shortened. 


190 


FIELD-MANUAL   FOR   ENGINEERS. 


Second  Solution.— Suppose  the  curve  run  backward  to  D  where 
the  tangent  is  parallel  to  BK,  and  draw  lines  as  shown.     Then 

DC  =  R  vers  0;    PM  =  r  vers  0. 


Hence     AE  -f  PM  =  AH  =  b  = 

But  Da  =  R  vers  ACD  =  R  vers  .#. 


If  r  =  R, 


+  Da-. 


+  «•  vers  _ZT=  5,  or  .4^  =  (b  -  r  vers  K  ) 


R 


Now  measure  ^.^  as  found,  making  angle  GAL  =  JfT.  Run 
^P  perpendicular  to  AE,  ar.d  the  carve  J.Pto  intersect  _Z£Pin  P. 
Reverse  and  run  the  curve  PB.  This  method  does  not  require 
the  central  angles  to  be  known,  and  neither  method  "equires  the 
curve  to  be  actually  run  backward  to  D.  The  problem  is  thus 
much  shortened. 

(6)  Receding   from  the   intersection   of  tangents.     (Fig.  110.) 


Given  the  angle  of  intersection  AKB  =  GAL  —  K,  the  distance 
AK  —  a,  and  the  radii  CA  =  72  and  OB  =  r,  to  find  the  central 
angles  G  and  0  of  the  reversed  curve  APB. 


REVERSED  CURVES.  191 

Draw  Aa  parallel  to  BK.     Then 

Da  =  R  vers  K-,    AH  =  AST  sin  K\ 

Da  +  Aff  =  DE=  &; 

Otf=  OB  +  DC  -  ED  =  R  +  r  -  b. 

r  -b  b 


•'     COS°=  r 

Evidently,  C  —  K  +  0. 

The  solution  here  given  is,  of  course,  applicable  to  the  former 
case  (Fig.  109)  ;  and  the  first  solution  there  given  is  likewise 
applicable  here. 

Example.—  Let  K  =  18°  23',  AK  =  920,  R  =  955,  and  r  =  819. 

Da  =  R  vers  K  =  955  X  .05103  =    48.73 
AH=  AKs\nK=  920  X  .31537  =  290.14 


338.87 

338.87--  1774  =  .19101. 
.-.     0  =  36°  00' 
K  =  18°  23' 


C  =  54°  23' 
Second  Solution. 

DC  —  R  vers  DCP  —  R  vers  0;     PM  —  r  vers  0, 
.-.  PM^Dc-  =  Da^  +  AE~. 

Hence         ^  JE'  +  PJf  =  AH  =b  =  AE(^-^^\  -f  Da~. 
But  Da  =  72  vers  ACD  =  R  vers  K. 


B 

».       7?  AT?         - 

7'  —  R,  AIL  =  — 


192 


FIELD-MANUAL    FOR   EKOIKEEES. 


Now  run  AE  and  EP  to  meet  the  curve  ADP,  etc.,  as  in  a 
former  case. 

VI.  To  connect  two  given  points  on  two  diverging  tangents  by 
a  reversed  curve. 

Given  the  chord  AB  =  a,  the  angles  BAL  =  A  and  ABK  = 
B,  and  the  radius  AC  —  ft,  to  find  the  radius  BO  —  r  of  the  re- 
versed curve  APB.  (Fig.  111.) 


FIG.  111. 
Draw  CT  parallel  and  CM  and  ONT  perpendicular  to  AB. 

GT  =  MN  =AB-  AM-  BN  =  a  -  RsinA  -  r  sin  B, 
and         OT  =  CM -f  ON  —  It  cos  A  -f  r  cos  B. 

Also,  CO  -—  E  -f  r. 

. ..  (R  -f  r)2  =  (a  —  #sin  J.  —  r  sin  5)*  +  (#  cos  A  -f-  r  cos  £)2. 

Expanding  and  observing  that 

1  -  (sin  A  sin  B  +  cos  ^L  cos  B)  =  I  —  cos  (A—  B)  =  vers  (4—5) 
we  readily  find 

a  —  R  sin  A 


T  = 


73 

—  vers  (A  —  B)  +  sin  B 
a 


Compare  with  Henck  Problem  38. 


REVERSED    CURVES. 


193 


Example.— I*t  a  =  1500,  fl  =  1600,  A  =  18°,  and  B  =  6°,  to 
find  r.     Then 

±a  =  750 

tfsinA^  494.43 


255.57 

-  vers  (A-B)  =  ~  .02185  =  .02331 
a  1^ 

sin  B  =  .10453 

.12784 

r  =  255.57  •*•  .12784  =  1999.14. 

VII.  Having  located  a  reversed  curve,  it  is  required  to  shift  the 
P.R.C.  so  that  the  termi-  ~ 

^1 

nal    branch   may   end   in  a 
tangent  parallel  to  the  orig-  \ 
inal  tangent.     (Fig.  112.) 

Let  APB  be  the  curve, 
CP=R,  OP=r,  and  BK~ 
D,  the  distance  »;; tween  the 
original  tangent  BL  and 
the  new  tangent  B'K,  to 
find  the  angle  of  retreat 
PGP'. 

Draw  Cmn  parallel  to  BL. 

We  have 


0 


FIG.  112. 


Oim  —  OiB'  +  B'H-\-  Hm  =  r  +  D  +  Hm, 
and  On  =  OB  +  Hm  '=  r  +  Hm; 

.-.   Oim  —  On  =  D. 

That  is,  (R  +  ?•)  cos  Oi-(R+  r)  cos  0  —  D; 

D 

.  •.  cos  0i  =  cos  0  -\-  -7,    ,      . 


Then  06*0,  =  A  =  0,Cn  -  OCn 

-  (90°  -  0.)  -  (90°  -  0)  =  0-  Oi. 

Example.- -Let  0  =_34°  20',  D  =  94,  7vJ  =  1433,  and  r  =  879, 


94  FIELD-MANUAL    FOR    ENGINEERS. 

We  have 

cos  0  =  .82577 
D-t-(K  +  r)  =  .04174 
.-.  cos  0,  -  .86751 
.-.  0,  =29°  50'; 
.'.  A  =  0  —  Oi  =  4°  30'. 


CHAPTER  VIII. 


TURNOUTS. 

A  TURNOUT  is  a  curved  track  leading  out  from  a  main  track. 
At  the  place  where  the  outer  rail  or  "lead"  crosses  the  rail  of 
the  main  track  a  frog  is  placed,  which  allows 
the  flanges  of  the  car-wheels  to  pass  the  rails. 

A  frog  is  a  piece  of  metal,  usually  steel, 
having  two  straight  channels  crossing  each 
other  on  the  upper  surface  through  which  the 
flanges  of  the  wheels  pass.  The  triangular 
part  of  the  upper  surface  formed  by  the 
channels  is  called  the  tongue  of  the  frog 
(Fig.  113).  Every  railway  company  should 
own  a  variety  of  frogs  suited  to  different 
turnouts  that  are  likely  to  be  required. 

Frogs  are  usually  designated  by  numbers, 
the  number  of  a,  frog  being  the  number  of 
times  the  bisecting  line  CF  contains  the  base 

flffl 

line   AB.     Denoting  the  number  by   n,  we  have   11  =  -— =,  and 
riff 

2n  =  -— .     Designating  the  frog  angle  AFB  by  F,  we  have  •• 


cot    F  = 


GF 
AC' 


=  2n 


(1) 


Standard  frogs  are  made  so  that  n  is  a  whole  number  or  a 
whole  number  plus  ^,  and  therefore  2n  is  a  whole  number. 

Let  Fig.  114  represent  a  "  split-switch  "  turnout.  AB  and  JIF 
represent  the  rails  of  the  main  track,  and  AC  one  of  the  switch- 
rails.  In  the  figure  the  switch  is  set  for  the  turnout,  but  when 
set  for  the  main  track  A  takes  the  position  of  A'. 

Prolong  the  curve  backward  to  c,  where  the  radius  cLO  is  per- 
pendicular to  the  main  track, 

195 


196 


FIELD-MANUAL  FOE  ENGINEERS. 


Draw  also  the  radii  GO  and  FO  ;  Ca  parallel  to  the  main  track  ; 
and  the  tangents  (7F7T  and  FV. 

Then  LFV—LOF  —  the  frog  angle,  which  we  will  denote  by  F. 
Let  8  =  the  switch  angle  BAG  =  FKV  =  LOG,  then 


COF  = 


-F-  8. 


Then 

Also,  DFC  =  LFV  -  CFV  =  F  - 


-  8)  =  %(F  + 


FIG.  114. 

Or,  since  the  tangent  CV  makes  an  angle  8,  and  the  tangent 
FVa.ii  angle  F  with  the  main  track, 

angle  FVK  =  COF  =  F  -  8. 

Hence  CFV  =  %(F  -  S), 

and    DFC  =  DFV  -CFV  =  F  -  %(F  -  8)  =  §(F  +  8). 

Let  R  =  R  +  \g  =  CO  =  the  radius  of  the  outer  or   "lead" 
rail,  and  If  —  the  degree  of  the  curve  CF. 

Let  d  =  A  A'  —  BC,  which  is  properly  the  throw  of  the  switch, 
and  I  =  the  length  of  the  switch-rail. 

Suppose  d  =  5  inches  =  .416  of  a  foot,  and  I  =  15  feet. 


Then 


TURNOUTS.  197 

.'.  S  —  1°  35'  30",     and     vers  S  —  .0003858. 
Now  cL  =-  R  vers  F,     and     ca  =  R  vers  8. 

.'.    R'  vers  F  —  R  vers  8  =  cL  —  ca  —  aL  =  CD  =  g  —  d. 


vers  F  —  vers  S      cos  S  —  cos  V 
or  vers  F  =  vers  S  -f-  : — —f — (2) 

We  observe  that 
vers  F  —  vers  S  =  (1  —  cos  F)  —  (1  —  cos  8)  =  cos  S  —  cos  F, 

so  that  in  all  such  cases  either  the  cosines  or  the  versines  may  be 
used  as  is  most  convenient. 


DF  =  CD  cot  CFD  =  (fj  -  d)  cot  i(F  -\-  8),    .     (4) 

HF=l  +  DF=l  +  (<?  -  d)cot$(F+8),      .     (5) 

and  Zj^  =  FO  sin  ZOF  =  ^'  sin  F.       .....     (6) 

HF  is  the  "  lead  "  and  shows  the  distance  apart,  measured  paral- 
lel to  the  main  track,  of  the  head-block  and  the  point  of  the  frog. 
Since  D'  represents  the  degree  of  the  curve  CF,  we  have 

F  —  8 
arc  CF=IW 


. 

The  lead  can  be  lengthened  or  shortened  somewhat  without 
materially  damaging  the  turnout  curve  ;  and  therefore,  in  prac- 
tice, rails  of  such  lengths  are  selected,  when  possible,  as  will 
make  the  lead  sufficiently  near  the  true  length,  thus  avoiding  the 
cutting  of  rails. 

In  case  of  sharp  turnouts  (those  of  short  radii),  point  or  switch 
rails  shorter  than  15  feet  should  be  used;  but  in  any  case  to  find 
the  radius  or  the  lead,  etc.,  it  is  only  necessary  to  substitute  in 
the  above  equations  the  proper  values  of  d  and  S. 

Table  VI  gives  the  elements  of  turnouts  for  a  variety  of  frogs, 
for  switch-rails  of  15  feet  in  length,  and  for  a  throw  of  5  inches  ; 


198  FIELD-MANUAL   FOB  ENGINEERS. 

and  also  for  a  Dumber  of  frogs  for  tlie  same  throw  and  for  switeh- 
rails  8  feet  in  length. 

Tlie  same  table  gives  also,  in  case  of  a  double  turnout  on 
opposite  sides  of  a  straight  track,  the  number  (n'),  angle  (F"), 
and  position  of  the  middle  frog,  corresponding  to  given  turnout 
frogs  F  and  F',  as  shown  in  Fig.  122.  The  similar  triangles 
ABC  and  aOO  of  Fig.  114  give 

A  f  7 

00  =  aC^,     or     R=aC-. 
BC  d 

It  in  this  we  put  aC  =  I,  we  have 


T.i  is  shows  the  value  of  R  such  that  the  end  c  of  the  prolonged 
curve  Cc  will  be  opposite  the  head-block  A, 

If  1=15  feet  and  d  ~  5  inches,  we  have  72  =  540.  And  if 
I  =  8  feet  and  d  =  5  inches,  2{  =  153.6,  Hence,  in  Table  VI,  for 
switch-rail  15  feet 

LF  ^  HF  accordingly  as  R  ^  540, 
and  for  switch-rail  8  feet 

LF  >  HF  accordingly  as  R  >  153.6. 

Example.—  Find  the  lead  and  radius  for  a  number  9  frog,  switch- 
rail  15  feet  and  throw  5  inches.  We  have 

F  =  69  21'  35",    and    8  =  1°  35'  30". 
.-,     vers  F  -  vers  S  =  ,005768,     and    g  -  d  =  4.2917. 
4  2Q1  7 


and 

jjp>=l  +  (g.,  d)  cot  #F+  8)  =  15  +  4.2917  X  14.3885  =  76.75. 


A  turnout  curve  between  head-block  and  frog  is  in  general 
not  the  same  as  the  main  part  of  the  curve  beyond  the  frog. 
When,  therefore^  two  intersecting  tracks  or  a  straight  line  and 


TURNOUTS. 


199 


track,  for  example,  are  to  be  connected  by  a  compound  curve,  the 
tangent  points  must  be  found  so  tbat  witb  given  frogs  tbe  con- 
necting curve  may  be  properly  located. 

Problem.  —  To  connect  a  straigbt  track  and  straight  line  by  a 
compound  curve  of  two  branches, 
the  first  branch  being  a  turnout 
curve  fitting  a  given  frog.  Eqs. 
(1)  to  (5)  of  compound  curves 
apply  to  this  case  without  change. 

Let  A  V  represent  the  center  line 
of  the  track,  BV  the  straight  line, 
and  APB  the  center  line  of  the 
compound  curve. 

Example.—  Let  AOV  —  Uv  =  437 
to  fit  a  No.  7  frog,  11*  =  BO*  =  537, 
F  -  50°,  and  Oi  =  10°,  which  is 
sufficient  to  clear  the  frog.  It  is 
required  to  find  the  tangent  point  A. 

Eq.  (4)  gives 


A  V  -  T  - 
A*  '- 


sin  V 
437  X- 357  4- 100  X  .234 


.766 


=  234.2. 


The  tangent  VB  is  not  generally  needed,  but  may  be  found  in 
the  same  way  from  eq.  (5), 

Problem. — To  connect  two  intersecting  straight  tracks  by  a  com- 
pound curve  of  three  branches,  the  two  turnout  curves  having 
radii  to  suit  given  frogs. 

Let  Fig.  116  represent  the  case,  APi  and  BP*  being  the  center 
lines  of  the  turnout  curves.  The  tangents  are  required. 

First  Method. — The  equation  under  Fig.  102,  compound  CUIVLS, 
applies  directly,  and  we  have 


BVs'in  V=(Ri  —  Z^2)  vers  0j-f-(7?2  —  R3)  vers(0!4-02 

4-  R*  vers  (0,  4  02  4-  0.), 
or,    j^Fsin  V=Ri  vers  Oi4-7?a[vers(0,4-0a)  — versO,] 

+  JR3[vers(Oi  4-  02  +  03)  -  vers(0i  4-  Oa) 


200  FIELD-MANUAL   FOK   ENGINEERS. 

To  find  A  V  we  may  interchange  7^  and  R* ,  also  Oi  and  03  in 
the  above  formulas,  as  explained  under  Fig.  102,  p.  177.  Thus, 
observing  that  Oi  -{-  0^  -}-  03  —  V,  we  have 


V—  (#3  -  P2)  vers  03  +  (R>  -  J2.)  vers  (0a  +  03) 

+  /?,  vers  F, 
and 

.4  Fsin  F=  jf?3  vers  03  +  7?a[vers  (Oa  +  03)  -  vers  0*] 

-  vers  (02  +  08)]. 


Example  1.—  Let  A01P1  =  14°  30',  Pi02P2  =  31°,  and  Py03B  = 
14°  30'.  Also  A0t  =  437  (to  fit  a  number  7  frog).  P202  =  490,  and 
P203  —  400.  To  find  AV.  Substituting  in  first  of  the  above 
equations,  we  have 

A  Fsin  F=  -  90  X  .03185  +  53  X  .29909  +  437  X  •§ 
=  -  2.87+15.85  +  218.5  =  231.48; 
~  231,48  -t-  ,886  ==  3C6.15, 


TURNOUTS. 


201 


Example  2. — Let  the  values  be  the  same  as  in  example  1,  ex- 
cept Hi  ,  which  we  will  suppose  equal  to  J?3  =  400.     Then 


_  Ry  =  —  90,     and 


—  R3  =  90. 


Therefore 


AVsin  V=  90(.  29909  -  .03185)  -f  400  X  .500 
.    =  90  X  .26724  -f-  200  =  224.05. 


Therefore 


A  V  =  224.05  •+- 


=  258.72. 


Eqs.  (a)  apply  without  change  whatever  may  be  the  order  of 
the  radii,  but  the  first  form,  which  usually  involves  the  least 
labor,  may  contain  some  negative  terms  which  must  be  subtracted 
from  the  sum  of  the  positive  terms. 

Second  Method. — Let  A'PiP9B'  (Fig.  117)  represent  a  symmetri- 
cal compound  curve.  Pro- 
long the  middle  arc'PjPa  to 
A  and  B  as  shown.  We  will 
illustrate  the  method  by  the 
following 

Example. — Let  the  vertex 
angle  =  V  =  98°,  A'0,Pt  = 
B'OiP*  =  9°  50',  which  is 


sufficient  to  clear  the  frogs.  .' 
Let  A'Oi  =  B'O,  =  72,  = 


437.06  corresponding  to 
number  7  frogs,  and  P,  0%  = 
/2,  =  358.1.  To  find  the  tan- 
gents  VA'  and  VB'. 

We  have,  from  eqs.  (7), 
(8),  etc.,  of  compound 
curves, 


AH=  (R,  -  #,)  vers  0,  =  79  X  .01469  =    1.16  =  o; 
A'H  =  (R,  -  l?a)  sin  0,     =  79  X  .17078  =  13.49  =  d. 


202  FiELb-MAJSTtTAL   FOR  ENGINEERS. 

.'.  HO*  =  R=  358.1  +  1.16  =  359.26  =  radius  of  simple  curve 
joining  IT  and  K,  but  not  used  except  to  find  VH  and  VA' . 

Now         VH=  R tan  £  V=  413.29,     and     VA'  =  426.78. 

This  gives  the  tangent  point  A'  (or  Z>')  from  which  the  com- 
pound curve  may  be  located.  The  simple  curve  joining  77 and  K 
is  not  shown. 

Let  Fig.  118  represent  the  case  in  which  the  curve  is  sharpened 
at  the  ends. 


Let  V=  84°,  A'OiPi  =  B'0,PA  =  29°,    A'0,  =  Rt  =  437.06, 
and  P,  02  =  ft*  =  877.06. 

Then     AH=(K*  -  #,)  vers  0,  =440  X  .12538  =    55.16; 
A'J7  =  (J*2  -  #0  sin  0!    =  440  X  .48481  =  213.32. 

.-.  H0y  =  7?  =  877.06  -  55.16  =  821.9. 
Now       VII  =  « tan  £  F  =  740.04. 

.-.   KA'  =  740.04  -  213.32  =  526.72. 

The  curve  A'PiPiB'  may  now  be  located. 

The  simple  curve  connecting  the  tangent  points  //and  /Tis  not 
drawn. 


TURNOUTS. 


203 


Third  Method. — We  may  easily  fit  a  given  turnout  curve  to  a 
different  curve  beyond 
the  frog  by  a  simple  but 
direct  application  of  Fig. 
81  and  corresponding 
formulas  of  compound 
curves. 

Thus  let  A'm'B',  Fig. 
119,  be  a  simple  curve 
of  radius  A'O  —  R  con- 
necting the  center  lines, 
A'V  and  B'V,  of  two 
tracks  ;  to  substitute  for 
it  a  compound  curve 
A'mB'  of  two  different 
radii,  A'O^-B'O^-  R, 
and  7>1Oa=A'fl, the  longer 
of  which  is  to  be  used 
for  a  distance  adjacent 
to  each  tangent,  for  the 
purpose  of  fitting  a 

given  frog.     The  tangents  and  the  elements  of  the  curves  A 
and  B'Pi  are  given. 

Example.— Let  V  —  98°,  and  A'V=  B'V  =  4.6.78. 

Then  R  =  A'O  =  BO'  —.  A'  Fcot  £  V  =  426.78  X  .8693  =  371. 
.-.  D  =  15°  26' 40". 

Let  A  Pi  and  B'P*  be  13°  06'  34"  curves  for  number  7  frogs. 
Then  A'0l  =  B'0l  =  R,  =  437.06,  and  Rl  -  R  =  66.06. 
Suppose  A' Pi  =  B'P*  =  75  feet,  then 

0,  =  |(13°  06'  34")  =  9°  50'. 
Eq.  (17)  p.  142  gives 


FIG.  119. 


=  66.06  X  1.193  =  78.94. 
7?,  =  437.06  -  78.94  =  358.12.     .-.  Z>2  =  16°. 
iP^Br,  is  the  same  as  A'PiPy 


Observe  that  this  curve, 
Fig.  117  or  of  Fig.  118. 


of 


204 


FIELD-MANUAL  FOR  KXGIXKERS. 


We  observe  that  the  degrees  of  the  two  branches  of  ihe  com- 
pound curve  (taking  A'P,  and  Ji'l\  together  as  one)  differ 
approximately  from  the  degree  of  the  simple  curve  inversely  as 
their  lengths. 

It  is,  of  course,  not  necessary  to  locate  any  simple  curve,  but 
simply  to  make  use  in  the  formulas  of  the  radius  (A'O,  for  ex- 
ample) of  a  simple  curve  corresponding  to  any  desired  tangent 
A'V. 

If  we  suppose  the  switch-rail  to  be  a  part  of  the  turnout  curve, 
.  such  curve  being  tangent  to  the  main 
track  at  A  (Fig.  120),  we  have 


AO"  = 


AH 


\ersAO"F' 

and  representing  A  0"  —  FO"  by  11"  to 
distinguish  it  from  the  true  radius  R'  = 
OF  (Fig.  114),  given  by  eq.  (2),  we  have 


R"  = 


r-   -     -     •     (?) 


FIG.  120 


vers  F' 
The  triangle  FHO"  gives 

HF  =  II"  sin  F. (8) 

The  triangle  AFH  gives 

HF=gcot$F,    .     .,':.     .     .     .     .     (80 

and  AF-    .  ^ljr  •    ^V£  ^  &  v    .     .     (9) 

These  equations  may  be  deduced  at  once  from  those  for  Fig.  114 
by  putting  d  =  0,  8  =  0,  and  I  =  0,  as  they  are  in  this  case. 
Introducing  the  number  of  the  frog,  we  have 


(8") 


AFH  = 

and  HF  =  AH  cot  AFII  =  g  cot  \F  =  2gn. 

Or,  make  HA'  =  II A ;  then 

AFA'  =  2AFH  =  F, 


TtTRKOtTTS. 


205 


find  the  triangle  A  FA'  may  represent  the  frog,  and  we  have,  by 
definition, 


J1F       HF  „„      0 

n  —  -r-r,  —  -.-,     or    HF  =  2gn. 
AA'        *g 


(8") 


Again. 

JJF*  =  AH(2AO"  -  All)  =  2AII(AO"  -  $AH)  =  2glil.    (S'") 
Equating  values  of  HF9  from  (8")  and  (8"'),  we  find 

the  radius  of  the  center  line  of  turnout  =  7?,   —  2ff)i*.     .     (10) 
A  F*  =  7IF*  +  tf  =  4g*n*  -f  cf-  •     .:  AF  = 


The  foregoing  equations  are  not  applicable,  however,  to  straight 
switch-rails,  as  we  will  explain. 

Let  ACiF(Fig.  121)  represent  the  turnout  curve  actually  located 
under  the  supposition  that 
the  switch- rail  is  a  part  of 
the  turnout  curve,  which 
is  tangent  to  the  rail  of 
the  main  track  at  the 
head-block  A,  AC  being 
the  switch-rail.  Eq.  (10) 
gives,  for  a  number  9 
frog, 

72=2X4.708X81=762.7. 

The  tangent  offset  of  the 
curve  AC\F  from  the  rail 
A  V,  at  a  point  opposite 
C,  is 


=  1.77  inches. 

Hence  the  curve  AdF 
is  5-1.77  =  8.23  inches  °i 

from  C,  the  end  of  switch-  FIG.  121. 

rail.  But  the  real  turnout  curve  must  connect  (7  and  T^and  be  tan- 


-.MM)  FIELD-MANUAL   FOR   ENGINKKRS. 

gent  to  the  switch-Tail  and  frog.     Prolong  AC  to  meet  the  tan- 
FK  in  K. 

Now     IlF=2gn;     angle  FA  K  =  $F-8;     AFK  =  I  F. 

Also,        sin  ^A^  =  sin  (FAK  +  AFK)  =  sin  (F  -  8). 


.0.  . 

sin  (F  -  £)  sin  (F  —  8) 

For  a  number  9  frog,  switch-rail  15  feet  and  throw  5  inches,  we 
have 

AK  =  56.64,     or     CK  =  41.64  =  T*  ,     and     FK  —  28.30  =  T. 

Since  the  tangents  are  unequal  a  simple  curve  cannot  meet  the 
required  conditions,  and  we  will  suppose  a  compound  curve  of 
two  branches.  Since,  moreover,  Ihe  tangents  are  quite  unequal 
and  the  central  angles  small,  we  know  from  Chapter  VI,  eq.  (3), 
and  eqs.  (e).and  (/)  following,  that  the  radii  will  differ  very  much. 
We  will  suppose  the  central  angles  equal,  for  this  will  make  the 
radii  the  most  nearly  equal  possible,  as  pointed  out  in  Chapter  VI, 
and  will  result  in  the  best  possible  curve.  The  vertex  angle 

CKV  =  V  =  F  -  8  =  6°  21'  -  1°  35'  =  4°  46', 
and  therefore  0,  =  02  =  2°  23'. 

Eqs.  (1)  and  (2),  Chapter  VI,  give 

(70,  =  #!  =  1164.13,     and    TV),  =  #a  =  513.85. 

We  note  that  the  offset  from  the  tangent  AV  to  the  curve  AC\F 
is  5  inches  at  about  25  feet  from  A,  and  that  the  curve  has  the 
direction  of  the  switch-rail  at  about  21  feet  from  A.  By  moving 
the  switch-rail  forward  from  .4  (7  to  AiCi  about  8  feet,  making  A  C 
about  23  feet,  a  close  approximation  to  the  part  of  the  curve  C\F 
could  be  used.  This  would  make  the  lead  nearly  equal  to  the 
true  lead  given  in  Table  VI.  This  would  also  involve  some  com- 
promise, and  would  be  somewhat  troublesome  in  practice. 

Thus  is  proved  ihe  impossibility  of  locating  a  simple  curve  tan- 
gent to  a  switch-rail  and  to  a  frog  placed  in  the  position  deter- 
mined by  the  above  formulas. 

When  so  placed  the  result  is  a  turnout  difficult  and  even  dan- 


TURNOUTS.  207 

gerous  to  operate,  unless  the  trackman  is  able  to  put  in  by  the 
eye  a  compound  curve  answering  more  or  less  well  the  purpose 
of  the  turnout. 

To  LAY  OUT  THE  TURNOUT.     (Fig.  114.) 

When  it  can  be  done,  which  is  usually  the  case,  first  locate  the 
frog  with  reference  to  a  joint  in  the  main  track.  Knowing  the 
length  from  point  of  frog  to  either  end  of  it,  we  thus  find  and 
mark  the  position  of  point  of  frog,  which  is  Fin  Fig.  114. 

Measure  off  FH,  giving  place  of  head-block  so  as  to  avoid  cut- 
ting rails  if  possible. 

For  example,  a  number  9  frog  calls  for  a  lead  HF  =  76.75.  Sup- 
pose the  length  of  frog  from  its  point  to  the  switch  end,  which  is 
the  part  that  applies  on  the  "lead,"  is  3.35  feet,  the  switch-rail 
being  15  feet,  This  will  use  up  3.35  -f  15.0  =  18.35  feet,  and  we 
will  need  76.75  -  18.35  =  58.40  feet  of  rails.  We  may  well  use 
two  30-foot  rails,  making  FH  =  3.35  +  15  +  60  —  78.35.  The 
arc  CF  is  a  trifle  longer  than  DF  (about  0.1  for  a  No.  12  and  0.2 
for  a  No.  7  frog),  and  this  will  practically  allow  for  expansion 
and  therefore  no  allowance  need  be  made  in  measuring  FH. 
Note  that  FD  =  FH  —  I. 

Remembering  that  the  curve  CFis  tangent  to  the  switch-rail  at 
C,  the  curve  of  the  outer  rail  CF  may  be  located  by  any  methods 
heretofore  explained. 

Drawing  CF,  the  curve  CFis  easily  located  by  ordinates,  noting 
that  CF  =  DF  nearly,  and  that  radius  CO  —  R'  =  E  -f  %g. 

The  turnout  can  be  conveniently  located  by  setting  the  instru- 
ment at  C,  turning  off  from  the  line  of  the  track  the  angle  S,  which 
gives  the  tangent  CV,  and  locating  the  curve  of  the  outer  rail 
CF.  The  degree  D'  of  the  curve  CF,  corresponding  to  li'  =  R+lg, 
may  be  taken  from  Table  I. 

Or  set  at  ^and  turn  off  from  FD  the  angle  F,  which  gives  the 
tangent  FV,  and  run  the  curve  both  ways  from  F.  This  is  the 
handiest  and  best  method. 

We  may  locate  the  center  line  as  follows: 

Make  CE  perpendicular  to  AC,  equal  to  \g,  then  run  the  curve 
from  E,  with  radius  EO  —  R  given  in  Table  I,  the  tangent  at  E 
being,  of  course,  parallel  to  AC. 

Or  make  Fm  =  \g  and  perpendicular  to  FVt  that  is,  make  angle 
DFm  =  90°  —  F,  and  run  the  curve  both  ways  from  m,  the  tan- 
gent at  m  being  parallel  to  "FY. 


208 


FIELD-MANUAL   FOR   ENGINEERS. 


DOUBLE  TURNOUTS  FROM  A  STRAIGHT  TRACK. 

In  case  of  a  double  turnout  (Fig.  122)  three  frogs  are  required; 
the  frogs  F  and  F'  being  the  same  as  F  in  Fig.  114,  the  third 
frog  being  at  F",  where  the  lead-rails  cross  each  other. 


Fro.  122. 

Draw  the  line  II' F"  parallel  to  the  main  track;  then  it  is  plain 
that  one  half  the  frog-angle  F"  is  equal  to  the  angle  made  between 
fie  line  H'F"  and  a  tangent  at  F"  to  either  arc  AF  or  A'F'. 
Therefore  the  formulas  for  this  case  may  be  written  at  once  from 
eqs.  (2),  (3),  (4),  (5),  and  (6)  by  putting  CD'  =  \g  -  d  instead  of 
CD  =  g  —  d,  and  \F"  instead  of  F.  Hence 


— rfer—        -o>    or    vers  p™=  vers  S-f 
vers  47^   —vers  $* 


TURNOUTS.  209 


H'F"=l  +  D'F".    .     (13) 
L'F"=R'  smlF"  .......     (14) 

Table  VI  gives  the  numbers  (M").  angles  (F"),  and  distances 
f  rom  head-block  of  middle  frogs  corresponding  to  frogs  of  different 
numbers  at  2^  and  F'. 

Example.—  Let  the  frogs  at  F  &nd  F'  be  number  8|.     Then 

F  =  F'  =  6°  44',     and     vers  F  -  vers  8  =  .006511. 
Eq.  (2)  gives 


Then  (11)  gives 

vers  £7<"  =  .0003859  +  ~~~  =  .0035253. 

.-.  \¥"  =  4°  40'  26",     and    F"  =  9°  20'  52". 
From  (4), 

D'F"  =  (\g  -  d)  cot  %$F"  +  8}  =  1.9375  X  18.2709  =  35,3998, 
and 
H'F"  =  50.3998. 


12 

Finally,  »"  =  ^  cot  ^"  =  —  —  =  6.11. 

6 

Hence  a  number   6   frog  at   F",    the   curves   being    slightly 
flattened  at  the  frog  F",  will  answer  the  requirement. 

To  find  an  approximate  relation  between  JFand  F". 

Comparing  eqs.  (2)  and  (11),  omitting  d  and  vers  S  f  rom  both, 
we  have 

vers  \F"  =  $  vers  F,     or    4  vers  \F"  -  2  vers  F. 

But  versines  of  small  angles  are  nearly  as  the  squares  of  the 
angles,     Hence,  approximately, 

vers  F"  =  4  vers    F", 


210 


FIELD-MANUAL   FOR   ENGINEERS, 


These  equations  give 

vers  F"  =  2  vers  F,     or     (F")*  =  '2(F)\     or  F"  = 
If          F  =  F'  =  6°  44',     or     n  =  w'  =  8|, 


tlien         n"  -          =  6.01  =  6.0,  nearly,    or 
t/2 


"  =  9°  32',  nearly 


Hence   in    practice    take   U   from    Table   VI    opposite    F,    and 
F"  =  ^2Fas  nearly  as  possible. 

In  case  of  a  double  turnout,  where  no  frog  f"  is  at  hand  of  the 

number  given  by  eq.  (11)  or 

by  F"  =  |/2  F,  we  may 
locate  the  turnout  as  a  com- 
pound curve,  and  thus  fit 
the  frogs,  provided  \F"  is 
less  than  F. 

Let  OF"  =  OtF"  =  R', 
and  EF"  =  E^"  =  ML 
(Fig.  123.) 

R'  and  the  lead  H'F'  are 
found  as  in  the  preceding 
problem. 

We  observe  that  in  Fig. 
114  the  inclination  to  the 
main  track  of  the  arc  at  0 
is  8;  while  in  Fig.  123  the 
inclination  to  the  main  track 
of  the  arc  at  F"  is  \V"  . 

Moreover,    CD  =  g  —  d  ; 
while  F"K  =  %g.     Hence  it 
is  evident  that  we  can  write 
FIG.  123.  at  once  the  relation  betweei 

F"  and  F  from  the  relation  between  8  and  F  by  substituting 
\~F"  for  8,  and  \y  f  or  g  -  d      We  thus  find 


or    B  = 


.  (IS) 


Since  the  versine  of  the  angle  of  the  inclination  of  the  curve 
cCF&t  any  point  to  the  main  track  is  equal  to  the  distance  of 


TURNOUTS.  211 

that  point  below  c  divided  by  tbe  radius  R,  it  follows  that  the 
difference  between  the  versines  of  such  angles  at  any  two  points 
is  equal  to  the  difference  between  their  distances  below  c  (or  from 
the  rail  AS  or  DF)  divided  by  R>.  See  eqs.  (2)  and  (15).  Thus 
the  relation  between  the  versines  at  any  two  points  may  be  written 
at  once. 

When  F  =  \F"  ,  li\.  =00,  showing  that  the  turnout  beyond 
F"  is  the  straight  line  F"  L.  This  is  apparent  from  the  figure, 
since  the  angle  KLl"  =  LF"  M  —  %F". 

When  F  <  \~F"  ',  Rt  is  negative,  showing  that  the  turnout  must 
reverse  at  or  beyond  F"  in  order  to  cross  the  main  track  at  the 
required  angle. 

By  prolonging  the  arcs  F^'C  as  in  Fig.  114,  the  equations  for 
the  last  two  cases  may  readily  be  deduced  in  the  same  \vay  as 
were  those  for  Fig.  114. 

Example.—  F  =  6°  44',  F"  =  8°  48'. 

From  (11)  we  find 

~d  19875    -  756  45 

~ 


~  versfF"-versfl  ~  .0025613 
and  from  (13) 

H'F"  =  I  +  (i?  -  rf)  cot  \(\F"  +  £) 

=  15  +  1.9375  X  19.0550  =  51,9191. 
(15)  gives 


-    2'35416    -  595  96 
~  = 


and,  from  (4), 

KF  =  \g  cot  ^(F+  IF")  =  2.35416  X  11.2789  =  26.552. 


FF"  may  be  found  by  (3),  but  is  not  needed. 

Supposing  F"  <  2F,  then  we  have  approximately  as  follows  : 

When    F"  =  F  \/2,     Ri  =  R,    as     above    pointed    out    (see 
Fig.  122). 

When  F"  <  F  f/2,  R1  <  R',  as  shown  in  Fig.  123. 

When  F"  >  F  \/2,  R>  >  R'. 


2L2 


FIELD-MANUAL   1<OK  ENGINEERS. 


TURNOUTS  FROM  CURVES. 

Given  the  radius  M  of  the  center  line  of  the  main  track,  and 
the  frog  angle  F,  to  determine  the  position  of  the  frog  and  the 
radius  R'  of  the  center  line  of  the  turnout. 

I.     When  the  turnout  is  from  the  inside  of  the  curve.  (Fig.  124. ) 
Let  AB  and  DF  be  the  rails  of  the 
main  track,  AC  the  switch-rail,  the 
arc  CF  the  outer  rail  of  the  turnout, 
and  .Ftlie  point  of  the  frog. 

Since  the  radii  FE  and  FK  are 
perpendicular  to  the  tangents  of  the 
two  curves  at  F,  the  angle  EFK—F. 
Complete  the  semicircle  DFL,  join 
FL,  and  dra,\\DH  parallel  to  it.  Let 
FCK=  C.  Then 


and 


ECF  =  EFC  =  C  +  S, 
CFK=  C+F  +  S. 


and 


Let    DKF  =  K     Then 
KLF  =  KFL  =  CDII  =  \K. 
Now 

DFK  +  CFD  =  CFK  =  C+F+  S, 
FDK  -  CFD  -  DFK  -  CFD  =  DCF  =  G. 


Subtracting  and  dividing  by  2  gives 

CFD  = 
We  have 


=,     and 


cotpr 


LF 
DH 


CL 
CD 


_  ?^ZJ? 

(j  —  d 

2K  —  d 


or 


Again, 


g  —  d 

_  CD  sin  CDF  _  (#-(/)  cos -pT 
•-•  ''lj'"       ~  ~$w*(F  I  8)  ' 


(18) 


TURNOUTS. 


213 


Also, 

CMF  =  F+  K,    and     CEF  =  CMF  -  ECM  =  F  -f  JiT -  S, 
and  we  have 

j  /'1T7I 

''-JT — w    \   -  (20) 

K.  —  o) 


II.   If  the  turnout  is  from  the  outside  of  the  curve  (Fig.  125), 
complete   the   semicircle,    join   FL' 
and  draw  DH  parallel  to  it  to  meet 


FC  prolonged. 
Now  EFK'  =  F. 
Let       DCF  =  C. 
Then    ECF  =  EFC  = 
.-.  (7F^'  = 


K' 


S. 


and 

J)FK'  -  CFD  =  FDL  -  CFD  ^  C. 

Subtracting,   and   dividing  by   2, 
gives 

CFD  -^ 

Let  DKF  -  K. 
Then 


KL'F=KFL'  = 
We  have 

cot  4/T  =  —4,     and    tan 


FIG.  125. 


S}  =  ^F* 

_OL;  _2JR+_^ 

'-CD~   g-d' 

cot  i^T  =  tan  %(F  +  Sf- 


Then 


_ 


sin  VFD 


(g  —  d)  cos  ^K 
siu$(F+  8)  '     ' 


.    (18') 
(190 


214 


FIELD-MANUAL   POK    ENGINEEKS. 


Again,       CEF  =  CLF  -  ECL,     and     CLF  =  F  -  K. 

.  •.  CEF  =  F  -  K  -  8. 
Finally, 

.  T3/     i      j       TT7//7 


-  K  -  S) 


Since  in  both  cases  jfiTis  small,  cos  ^K  =  1,  very  nearly,  and  the 

lead  =  - — ^—fj 5-,  very  nearly,  which  is  the  value  of  the  lead 

sin  •%(!*  -j-  o) 

in  a  turnout  from  a  straight  track.     Hence  Table  VI  applies  to 
turnouts  from  curves  as  well  as  from  straight  lines. 

In  case  of  a  double  turnout  on  opposite  sides  of  a  curve,  Fig. 
126,  the  frogs  F  and  F',  if  equal,  are 
opposite,  since  the  chords  AF'  and 
A'F,  as  we  have  just  seen,  are  very 
nearly  equal. 

Comparing  Figs.  122  and  126,  ig- 
noring vers  S  and  d,  we  see  that,  in 
both  cases,  F  '  is  in  the  center  of  the 
main  track,  and  that  the  relation  be- 
tween F  (or  F')  and  F"  is  the  same 
in  this  case  as  in  the  case  of  a 

h[ -~^XF^<L  double  turnout  on  opposite  sides  of 

^1— =zd__   \  r"--^    ^\        a  straight  track;  and  consequently, 
in  this  case  also,  we  have 

vers  \T?"  —  \  vers  F, 
vers    F"  =  2  vers  F, 

F"  —  \/2F,  nearly. 

We  learn  from  the  preceding  dis- 
cussion and  results  that  frogs  in  the 

77 '" 
approximate  ratio  of     ^  =  |/2  are 

needed  in  the  cases  of  double  turn- 
outs on  opposite  sides  of  a  straight 
FlG>  126.  truck  or  on  opposite  sides  of  a  curved 

.track.     Such  frogs  should  therefore  be  kept  on  hand. 


TURNOUTS.  215 

Thus  if  F  corresponds  to  a  number  8|  frog,  then  F"  corre- 
sponds to  a  number  6  frog1,  nearly. 

Problem.  —  To  rind  an  approximate  value  of  the  degree  of  a 
turnout  from  a  curve  in  terms  of  the  degree  of  a  turnout  from 
a  straight  line,  the  turnouts  to  fit  the  same  frog,  F. 

Let  d  =  the  degrete  of  the  center  line  of  the  main  track,  D  ~  the 
degree  of  the  center  line  of  the  turnout  from  the  main  track,  and 
D\  =  the  degree  of  the  center  line  of  the  turnout  from  a  straight 
track  to  fit  the  frog  F,  as  in  Fig.  114. 

Let  I  =.  the  length  of  either  arc  CF  or  DF  in  Figs.  114,  124, 
and  125.  Ignoring  the  difference  in  curvature  between  the  center 
line  and  either  rail  of  a  track,  we  have  the  following  equations  : 

In  Fig.  114  the  difference  in  the  directions  of  the  arc  at  C  and 

at  F  is  evidently  equal  to  F  —  8.     But  it  is  also  equal  to  r7^D\- 
Hence  ^D,  =  F  -  8.       ......     (21) 

In  Fig.  124,  since  the  arcs  CF  and  DFwake  an  angle  of  8  with 
each  other  at  C  and  D,  and  an  angle  of  F  with  each  other  at  F,  it 
is  evident  that  the  change  in  direction  of  the  arc  CF  from  C  to  F, 
less  the  change  in  direction  of  the  arc  DFfrorn  D  to  F,  is  equal 

to  F  -  8.     But  the  change  in  CF=  -^D,  and  that  of  DF=  ~d. 

100  100 

Hence  D~d=F-s- 


Equating  with  the  above  and  dividing  by  —  —  ,  we  find 

100 


(22) 


In  Fig  125  the  change  in  direction  from  C  to  F  plus  the  change 
f  rom  D  to  F  is  equal  to  F  —  8.     Hence 

I  I 

mD~^"md  =  F~s' 

Equating,  we  find 


216 


FIELD-MANUAL   FOR   ENGINEEKS. 


Hence,  to  find  the  degree  of  a  turnout  for  a  given  frog  from  the 
inside  of  a  curve:  To  the  degree  of  a  Turnout  for  the  same  frog 
from  a  straight  track,  add  the  degree  of  the  yiccn  track. 

To  find  the  degree  of  a  turnout  for  a  given  frog  from  the 
outside  of- a  curve:  From  the  degree  of  a  turnout  for  the 
same  frog  from  a  straight  track,  subtract  "the  degree  of  the 
given  track. 

Example,  1. — Required  the  degree  of  a  turnout  from  the  inside, 
as  well  as  from  the  outside,  of  a  4°  curve  for  a  No.  9  frog. 

The  degree  of  a  turnout  from  a  straight  track  for  a  No.  9  frog 
is  7°  44'.  Hence  the  turnout  from  the  inside  is  11°  44',  and  from 
the  outside  is  3°  44'.  (See  Figs.  124  and  125  ) 

Example  2. — Required  the  degree  of  a  turnout  from  the  outside 
of  a  7°  44'  curve  for  a  No.  9  frog. 

In  this  case  the  degree  of  the  turnout  is  7°  44'  less  7°  44',  or  0. 
Hence  the  turnout  is  a  straight  track.  In  this  case  the  turnout 
may  be  regarded  as  the  main  track,  and  the  main  track  as  the 
turnout.  (See  Fig.  114.) 

Example  3. — Required  the  degree  of  a  turnout  from  the  outside 
of  a  11°  curve  for  a  No.  10  frog. 

In  this  case  the  degree  of  the  main 
track  exceeds  the  degree  of  a  turnout 
from  a  straight  track  for  the  given 
frog  by  11°  -  (6°  09')  =  4°  51'. 
Hence,  referring  to  Fig.  127,  we  see 
that  in  this  case  the  center  Ei  of  the 
turnout,  instead  of  being  on  the 
outside  of  the  main  track  as  in  Ex- 
ample 1,  is  on  the  inside,  that  is,  on 
the  same  side  as  K. 

Example  4.— Let  F  (in  Fig.  124) 
=  5°  43'.  Main  track  an  8°  curve, 
or  r  =  716.197.  5=1°  35'.  Re- 
quired the  lead  CF,  and  the  radius 
of  the  center  line  of  the  turnout. 
We  have 


FIG.  127. 


CF=  v-^~ 


4  2916 


The  degree  of  turnout  curve  from  a  straight  track,  correspond- 
ing to  the  frpg  F,  is,  by  Table  VI,  6°  09'  -J-. 


TURNOUTS. 


Adding  the  degree  of  the  main  track  (8°),  we  have 

degree  of  turnout  =  14°  09'  +» 
and  the  radius  is  It  =  404.92  — . 

Or  (18)  gives 

(2R  —  d)  tan 


tan  KDF  — 

g  -  a 

.-.     KDF=8T1S' 
Then  (19)  gives 

GF  = 
Then,  by  (20),  we  find 


(g  —  d)  cos 


1431.98  X  .06879  _  . 
4.2916  =      ' 


=  67.34. 


33.67 


=  406.01, 


sin£(F+#-,8)  -  .08293 

and  11'  =  403.66,  and  D  =  14°  11'  +  . 

The  above  approximate  results  differ  but  little  from  the  true 
results  and  show  that  the  approximate  formulas  are  practically 
all-sufficient. 

OTHER  TURNOUTS  FROM  A  STRAIGHT  TRACK. 

From  the  same  point  in  a  straight  track  it  is  required  to  locate 
two  turnouts  on  the  same 
side.     (Fig.  128.) 

We  will  assume  that 
F=  F',  and  that  these 
frogs  are  opposite  to  each 
oilier,  and  that  the  curves 
are  tangent  to  the  main 
track  at  A. 

Let  KA  =  r  -f  %g,  and 
EA  =  r'  +  \g. 

The  angle  AKF=F,  and 
the  angle  KF'E-F'  -  F. 

Hence  the  triangle  KEF' 
is  isosceles.     Therefore 
EK  =  EF'  =  AE, 
or    AE=\AK. 
That  is.  FIG.  128. 


(24) 


£18  £tELt>-MAttUAL   FOR   ENGINEERS. 

This  relation  is  easily  proved  as  follows  : 

Let  d  =  the  degree  of  the  curve  AF,  and  d'  that  of  AF'.  Now 
we  have  seen  that  d'  =  d  -f-  the  degree  of  a  turnout  from  a 
straight  track  for  the  frog  F,  i.e.,  d'  =  d  -f  d,  or 

' 


Or  again  :   Draw  the  tangents  F7^'  and  DF  '';  also  777^'  parallel 
to  LF.     Evidently, 

VF'D  =  F'  =  F,     and    DF'H  =  F. 

.-.      VF'H=2F. 
Hence 

AH         AH 


vers  FF'lf  =  vers  AEF'  =  vers  2F  = 


AE      r'  —  %g' 
Also, 


•*•     **'  +  $#  =  Ur  +  %9\  as  before. 

In  this  we  assume  £77  =  ^IZ  =  F'F-  but  1/77  =  F'F  cos  Jf^. 
We  also  assume  vers  2F  =  4  vers  F ;  but  vers  27^  <  4  vers  7^. 
The  two  small  errors  exactly  balance,  however,  and  result  in  a 
true  formula. 

We  have  vers  F'  =  vers  F=  TT>      ....     (25) 


and  vers  F"  —  -^- 


But  r>  +  ft  =  fr  +  ig). 

.'.     vers  jp"'  =  2  vers  7^,     or     F"  =  V^F,  nearly. 

This  is  the  same  relation  between  F  and  F"  as  was  above 
shown  to  exist  between  the  frogs  in  the  case  of  turnouts  to  the 
opposite  sides  of  a  straight  or  of  a  curved  track.  (Figs.  122  and 


TURNOUTS. 


219 


126.)  Hence  a  set  of  frogs  adapted  to  a  double  turnout  on  oppo- 
site sides  of  a  straight  or  a  curved  track  is  also  adapted  to  sucli  a 
turnout  on  the  same  side  of  a  straight  track.  In  case  of  the 
double  turnout  on  the  same  side  of  a  straight  track,  the  longer 
radius  is  equal  to  twice  the  shorter  radius  -j~  \g. 

In  case  no  frog  is  at  hand  equal  to  F"  given  by  the  relation 
vers  F"  =  2  vers  F,  we  may  select 
one,   which   call  F",  as  near  the 
same  angle  as  possible,  and  find,  as 
already     shown     (see    Fig.    129), 


^"  =  ,'+k  =  _  and 

BF"  =  (r'  -f  |#)  sin  F".  Then 
compound  the  curve  at  F",  and 
find  the  radius  OF'  to  suit  the 
frog  F',  whether  it  is  equal  or  un- 
equal to  F. 

Let  AK  =  r  -f-  \g  as  before. 

Then    KF'  =  r  -  %g. 
Assume 

OF'  =  r"  +  \g.  FIG.  12&. 

Then  EO  —  r'  -  r",     and     BK  =  r  -  r'. 

BEF"  =  F",      and      OF'K  =  F'. 
Let  F'KF"  =  K.     We  have 

**>' 


BK  ~  r  -  \q  ..... 

Then     OF"K  =  BEF"  -  BKF"  =  F"  -  BKF"  1=  A, 
Again, 

F"F'K  =  F"F'0  +  OF'K=  F"F'0  -f  F', 
and         F'F"K  =  F'F"0  -  OF"K=  F"F'0  -  A, 
Subtracting  gives 

F"F'K  -  F'F"K  =  A  +  F'. 


(27) 


220  FIELD-MANUAL   FOR   ENGINEERS. 

Also, 

TUP" 
KF"  =  ^KF'"  and  UF"  =  KF"  ~(r~  W  =  ''  Say>  (28) 

Hence 
KF"  +  KF  =  2(r  -  $g)  -f  <?,    and    KF"  -  KF  -  e. 

Applying  the   "tangent  proportion  "  to  the  triangle  KF'F", 
we  have 

e  :  2(r  -  %g)  +  e  :  :  tan  \(A  +  F'}  :  cot  \K, 

or  cot  *  JT  =  P*  ~  ^  +  6>)  tan  \(A  +  F7).      .     •     (29) 

Now 

=  BKF"  +  iT,     and    5^"  =  2(r  -  ^)  sin  ^^F'.    (30) 


This  gives  the  position  of  F'. 
We  have 

\(KF'F"  +  KF"f")  =  90°  -  fZT, 

and          \(KF'F"  -  KF"F')  ~  ftA  +  F7)»  Just  found. 
Subtracting  gives 

^/?"'F'  =  90°  -  {(A  +  V  +  JT). 


Adding  OF"K  =  A  to  the  above,  we  have 

OF"F'  =--  90°  +  i-4  -  i(F'  -f  A"). 

Taking  OF"F'  +  OF'F"  —  20F''F'  from  180°,  we  have 
F'OF"  =  F'  +  K-  A. 

Finally, 


TURNOUTS.  221 

Suppose  the  turnout  straight  beyond  F",  the  frog  being  at  Flf 
In  the  triangle  Fi 


angle  F,V"R  =  90°  -  OF"K  =  90°  -  A. 

.-.     sin  FiF"K  —  cos  A. 
Also,  KFi  =  r  -  \g,     and    KF"  =  r  -  ig  +  e. 

Hence  sin  KF,F"  =  '' 


r  -  iff 
"  =  180°  -  (KFiF"  +  KF"F,\ 


Then       F"*  =  KF,  "=      ~        *™>.        (33) 

'  cos  A 


This  gives  the  position  of  the  frog  Ft. 

Example.—  F  =  6°  43'  59",  F'  =  6°  01'  32",  .F"  =  8°  47'  51". 

By  Table  VI,         BF  =  80.036,  r  =  680.306; 

£^"  =  61.204,  r'  =  397.826. 

By  (27), 

tan  BKF"  =  J^4  =  .0902777.  .'.     BKF"  =  5C  07'  31". 

o  M  .95^ 

vl  =  F"  -  BKF"  -  3°  38'  20";  A  -f  F'  =  9°  39'  52"; 

i(^.  -f  -F")  =  4°  49'  56". 
By  (28), 

fi1  904. 


~ 
.089913 

Eq.  (29)  gives 


=  680.702.        .-.     e-  680.702  -  677.952  =  2.75. 


>ot  lK=  tan 


-.     \K  =  I0  22'  42",     and  K  =  2°  45'  24". 
"^'  =  90°  -  £(4  +  17"  -f  7T)  =  83°  47'  22". 


£22  FIELD-MANUAL   FOR   ENGINEERS. 

Then,  by  (31), 

v,,v,       (r  ~  \g]  sin  K       677.952  X  .048943 
sin  KF"F'  -^94181- 

\(F'  +  K  -  A)  =  2°  34'  18". 
Then  (32)  gives 


and  r"  =  363.132. 


CHAPTER  IX. 
THE  TRUE  TRANSITION  CURVE. 

THE  object  of  tliis  chapter  is  to  make  known  the  true  transi- 
tion curve,  to  show  its  need,  to  furnish  sin, pic  foru.ulas  for  its 
use,  and  to  show  its  ready  application  in  practice. 

FUNDAMENTAL  PRINCIPLES. 

When  a  car  passes  from  a  straight  line  upon  a  curve,  or  vice 
versa,  it  receives  a  shock  more  or  less  severe,  in  proportion  to  the 
sharpness  of  curvature  and  the  rate  of  speed.  This,  it  is  well 
known,  is  due  to  a  tendency  of  any  body  in  motion  to  persist  in 
its  direction  of  motion  at  every  point  of  its  path. 

This  shock  is  damaging  to  rolling  stock  and  track,  causes  dis- 
comfort to  travelers,  and  is,  moreover,  a  source  of  danger. 

It  is  evident  that  the  elevation  of  the  outer  rail  cannot  obviate 
this  difficulty  to  any  appreciable  extent ;  for  it  is  plain  that  such 
elevation,  which  must  vary  directly  with  the  degree  of  curvature 
(see  Elevation  of  Rail,  Chap.  V),  can  correspond  to  a  gradual 
change  of  curvature  only,  and  not  to  a  sudden  change,  like  pass- 
ing from  a  straight  line  upon  a  curve  or  the  reverse. 

Whatever,  therefore,  the  elevation  might  be,  a  car  passing 
from  a  tangent  upon  a  sharp  curve,  or  conversely,  would  receive 
a  violent  shock.  The  only  way  possible  to  do  away  with  such 
shocks  and  the  resulting  evils  is  to  interpose  between  the  tangent 
and  the  main  curve  a  "transition  curve." 

A  transition  curve,  as  its  name  indicates,  is  a  curve  placed 
between  a  tangent  and  a  main  curve. 

A  theoretically  perfect  transition  curve  must  have  the  follow- 
ing properties  : 

1.  It  must  end  on  the  curve  with  a  radius  equal  to  that  of  the 
curve,  and  on  the  tangent  in  a  straight  line. 

2.  The  degrees  of  curvature  of  the  transition  curve  at  different 
points  must  vary  directly  as   their   distances   from  its  junction 
with  the  tangent. 


224 


FIELD-MANUAL    FOR    ENGINEERS. 


To  these  must  be  added  the  following  practical  requisites: 

3.  The  curve  should  be  one  easy  to  understand  and  especially 
easy  to  lay  out. 

4.  It  should  be  flexible,  so  as  to  accommodate  itself  to  the  con- 
figuration of  the  ground  and  to  other  conditions. 

Since  the  curvature  of  the  transition  curve  is  less  than  that  of 
the  main  curve  with  which  it  connects  except  at  and  near  the  point 
of  tangency,  it  must  lie  outside  of  the  latter  prolonged. 

Hence,  when  a  curve  is  located  ending  in  a  tangent,  it  is  neces- 
sary either  to  move  the  curve  inward  or  the  tangent  outward,  in 
order  to  interpose  a  transition  curve  between  them.  We  wrill 
suppose  the  curve  to  be  moved. 


Thus,  referring  to  Fig.  130,  AO  represents  a  tangent  and  A  the 
tangent  point  of  the  curve  Afi,  which  is  replaced  by  the  slightly 
sharper  concentric  curve  Ti.de.  This  latter  curve  is  connected  with 
the  original  tangent  at  0  by  the  transition  curve  Omd,  m  being 
the  intersection  of  that  curve  with  AK,  which  is  the  distance  be- 
•tween  the  concentric  curves,  and  is  called  the  offset. 

Since  the  curves  Kd  and  Omd  subtend  the  same  central  angle, 
while  the  latter  is  flatter  than  the  former  it  must  be  longer  also, 
and  therefore  0  must  be  back  of  A. 

Again,  since  the  radius  of  curvature  of  the  transition  curve  at 
id  is  equal  to  dE,  which  is  less  ih&nfS,  we  see  that  the  curvature 
of  that  curve  at  and  near  to  d  is  a  little  sharper  than  that  of  Hie 
•original  curve  Afi. 

Now  the  degrees  of  curvature  of  different  arcs  are  as  the  angles 
turned  in  passing  over  equal  lengths  of  those  arcs.  Hence,  in 


THE   TltrE   TRANSITION   CURVE.  225 

order  that  the  degrees  of  curvature  at  different  points  of  Omd 
ma.-  vary  as  their  distances  from  0,  the  angles  turned  in  passing 
over  arcs  of  the  same  length,  supposed  to  be  very  short,  must 
varv  as  the  distances  of  their  centers  from  0;  and  the  total  angles 
turned  between  0  and  different  points  must  therefore  vary  as  the 
squares  of  the  distances  of  those  points  from  0. 

Hence  if  0  is  the  total  angle  dgT turned  in  passing  over  any 
length  of  this  curve,  as  Omd  =  S,  we  must  have 

/S'2  =  cO,     c  being  a  constant (1) 

A  transition  curve  should  be  long  enough  to  allow  the  outer 
rail  to  gain  the  elevation  at  d,  proper  for  the  main  curve  without 
inclining  too  abruptly. 

Froude,  Rankine,  and  others  allow  an  incline  of  one  foot  in  300; 
and,  since  the  maximum  elevation  of  the  outer  rail  should  not 
exceed  8  inches  or  £  of  a  foot,  transition  curves  need  not  exceed 
$  x  300  =  200  feet  in  length.  Curves  of  any  length,  however, 
may  be  readily  used  by  the  formulas  to  be  given.* 

ELEMENTARY  RELATIONS. 

By  elementary  relations  is  meant,  those  pertaining  to  lines  and 
angles  of  the  figure. 

1.  To  find  the  length  of  the  offset  curve  in  terms  of  the  length 
of  the  transition  curve. 

Let  dEK  =  dgT  =  0,  and  8  =  the  length  of  the  transition 
curve  Omd.  Then 

Kd  =  |,   exactly (2) 

R 

The  figure  shows  that  md  is  a  trifle  greater  than  Kd  =   -,  and 

2 

o 

therefore  mO  is  a  trifle  less  than  Kd  =  -. 

QJ 

Also,  AO  is  a  trifle  less  than  mO,  hence  a  trifle  less  than  -. 

2 

Let  dE=  R',  then 

AT  =  dh  =  R'  sin  0 (3) 

*  The  author  hopes  to  give  the  proof  of  the  following  formulas  in  a  treat- 
ise on  the  True  Transition  Curve,  tq  appear  later. 


226  FIELD-MANUAL   FOR   ENGINEERS. 

2.  To  find  any  tangent  distance. 
We  find  Or=fl(l  _£  +  *._),    .....     (4) 

or  OT  =  S  cos  2~0  .........     (5) 


4 

or  OT  —  S  cos  -0,  very  nearly  .....  (6) 

9 

Example.—  Let  0  -  i  of  the  unit  angle  =  19°.09859.     Then  (4) 
gives 


OT=   i-+--  =5(1  -.011  +.  00005716-)  =.9889465. 


Eq.  (5)  gives  OT=  .9889365. 

Eq.  (6)  gives  OT=  .9890465. 

Hence  (5)  gives  result  too  small  by  only  .000015,  and  (6)  a  result 
too  large  by  only  .00015. 

0  iu  this  example  is  very  large  and  the  errors  are  therefore 
comparatively  very  large.  Hence  (6)  is  sufficiently  exact  for  all 
cases. 

Let  R>  =  KE,     Now  Kd  =  KB  X  KEd.     That  is, 


Again  we  find 

AO  —   -  /1  — _j_              )  (9\ 

U~   2\        30  +  1080~r 18J 


AO  =  -  cos  y^rO,  almost  exactly,      ...     (9) 

Q  1 

f  AO  =  -cos-0,  very  nearly (10) 

For  0  =  19°.  09859, 

(8)  gives  AO  =  |(1-.00370+.00001143)  =  .498155, 

(9)  gives  AO  —  .498155, 
and  (10)  gives  AQ  =  .498265. 


THE   TRUE   TRANSITION    CURVE.  227 

We  observe  that  (9),  even  for  this  very  large  value  of  0,  gives 
a  result  true  to  five  places;  and  that  (10)  is  also  practically  exact. 
Letting  KE  =  R',  we  also  find 

AO  -  fR'  sin  f  0  .......     (11) 

To  find  the  tangent  distance  corresponding  to  any  part  of  the 
curve  it  is  only  necessary  to  substitute  in  (4),  (5),  or  (6)  the  corre- 
sponding values  of  the  arc  and  of  tie  angle.  Thus  suppose  Gab 

=  $8.     Then  the  spiral  angle  for  6  is  —  ,  and  we  have 


OA'=li(l  ~  160  +  2565^6-)= 


C  -1 

=  g-  cos  -0,  very  nearly.       (13) 


No\*,  from  (10)  and  (13), 

O 

A  A'  =  OA'  -  OA  =  -^-(cos  £0  -  cos  J0).      .     (14) 
For  0  =  19°.09859, 


AA'  =  .00278?-=  .00139& 

A 

This  shows  the  distance  of  the  middle  of  the  curve  6  to  the  right 
of  AmK. 

For  all  ordinary  values  of  0  this  distance  can  be  neglected  and 
the  middle  of  the  curve  be  taken  as  at  m. 

From  (9)  and  (10)  we  have 

o  o 

—  -  AO  =  —  vers  ^0  =  d,  suppose,  .     .     .     (15) 

Q  Q 

and  -  —  AO  =  —  vers  £0    =  d,  suppose.    .     .     .     (16) 


These  values  are  very  easily  computed,  since  the  versines  of  the 
angles  contain  few  significant  figures. 


228  FIELD-MANUAL   FOR   ENGINEERS. 


o 

Then,  d  being  very  small,  AO  =  —  —  d  may  be  easily  computed 

4 

mentally. 

3.  To  find  any  offset,  as  dT.     We  find 


GO  ^ 

dT  =  —  sin  £  0  -j-  =  —  sin  $0,  very  nearly.    | 


Also,  dT—dVcosO. 

For  0  =  19°.09859,  (17)  gives 

dT=  .11033$ 
and  (18)  gives 


which  is  sufficiently  accurate  even  for  this  very  large  value  of  0. 

The  offset  for  any  point  of  the  curve  is  readily  found  by  sub- 
stituting in  (1?)  or  (18)  the  arc  and  the  angle  corresponding  to  the 
point. 

Thus  let  c  =  arc  Om,  and  E  =  the  corresponding  spiral  angle. 

Then  A»  =  «(f       **  +  ^n  -I  •    (19) 


9  BT 

or  Am  =  t  =  -^  sin  ~ (20) 

6  O 


Eq.  (1)  shows  that  E  varies  as  c2.     Also,  sin  \E  increases  less 
rapidly  than  E,  and  therefore  less  rapidly  than  c2. 
Hence  t  increases  less  rapidly  than  c3. 

8 

Again,  let  b  be  the  middle  point  of  the  curve.     Then  Ob  =  — , 

and  the  angle  subtended  by  the  arc  Oab  —  — . 


THE  TRUE  TRANSITION  CURVE.         229 

Substituting  these  values  for  S  and  0  in  (17),  or  for  c  and  E  in 
(19),  we  have 


Q  /^ 

or  -A'&  =  T  sin  —  ,  almost  exactly.    .     .     .     (22) 

4          o 

The  error  of  (22)  is  less  than  -.0000005  for  0  -  19°.09857. 
Eq.  (7)  gives 


0*f       O4        O6      \ 

0  =  -^  -  ^  +  ^  -J. 


Also, 


rr 

/i-fiT  =  —  sin  £  0,  very  nearly,     ......     (24) 

o 

7iA"  =  y\/S  sin  T7¥0,  almost  exactly  .....     (25) 
Now  from  (17)  and  (23)  we  get 


•  <26) 

Hence 

Q  S}  Q  /~l 

AK  -  ti  =  y  sin  —  +  =  —  sin  --,  very  nearly,      .      (27) 

or        AK  =  ti  =  T2T5sin  ^{0,  with  great  accuracy.  .     .     .     (28) 
Comparing  (18),  (22),  and  (27),  we  see  that 

A'b  >  iAK,     A'b  >  \dT,     and    AK  > 
However,  when  0  is  not  very  large,  we  have 


230  FIELD-MANUAL    FOR   ENGINEERS. 

A'b  =  \AK  =  IdT,  very  nearly  ;  |      .     .     .     (29) 
Am  =  \AK  —  \dT,  very  nearly.  )      .     .     .     (30) 

From  (10)  and  (28)  we  have 
|siniO    i 


+=    sm  -,  very  nearly.  (31) 


—  cos^O  —  cosiO 

The  above  values  are  all  very  simply  expressed  in  terms  of  S 
and  0,  the  offset  curve  being  known. 
We  now  require  the  following  : 
4.  Having  an  offset   t  corresponding  to  an  offset  curve  Kd,  of 

rr 

length  —-  and  degree  D,  to  find  the  change  in  the  offset,  or  the 
a 

new  offset,  t',  for  the  same  length  of  offset   curve,  when  the 
original  Afi  is  a  D  degree  curve. 

O  r\ 

We  have,  from  (28),  AK=  t  =  —  sin  —  ,  when  Kd  is  a  D  degree 

8 

curve  of  length  —-.  the  central  angle  being  0. 
a 

When,   however,   Afi  is  a  D  degree  curve,  Kd  is  a  curve  of 

A  W  7? 

degree  Dj^,  ==  D——*—f  —  D',    and,    its   length    remaining  the 

JiHi  41    —     t 

n 

same,  the  central  angle  =  0  =  -  /  =  0'  ',  and  the  offset  t'  >  t. 
H  —  t 

We  find 

'     (83) 


Or  we  may  find  t'  as  follows : 

Suppose,  for  a  moment,  Kd,  instead  of  Af,  to  be  a  D  degree 
curve  of  radius  R,  and  compute  t  or  take  it  from  Table  XV.  Then 
we  have  KE  =  R  —  t  —  R',  and  of  course  D',  with  which  we  may 
compute  t'  very  accurately,  or  we  can  take  it  from  Table  XV. 

5.  To  find  the  offset  A K  =  t  in  terms  of  the  central  angle  0 
and  the  radius  KE  =  R'  or  of  AE  =  R. 

Substituting  20R'  for  S  in  (27)  and  dividing  by  R'  gives 

AK  _  (0*  _    O4         0G 
~$~  ~  IT  "  108  +  4680 


THE   TRUE   TRANSITION   CURVE.  231 

Represent  the  series  by  e  ;  then 

-=e,    or    t  =  R'e.       .  '  .....     (34) 


Substituting  R  —  t  for  Rr  in  (34),  we  find 


(35) 


But          e  =  l  vers  f  0  -f  —  f  vers  f  0,  very  nearly.     .     .     (36) 
Hence,  from  (34)  and  (36), 

t  =  R$  vers  f  0,       .     .     .....     (37) 

and,  from  (35)  and  (36), 

vers    ° 


- 

~ 


f  +  vers  f  0 


6.  To  find  the  angle  between  the  tangent  AO  and  any  radius 
vector  or  chord  drawn  from  0. 
We  find 

dl  d  0        0s  0s 

~-  =-    + 


05       6000  ' 
representing  any  arc. 

.'.     AOd  <  —  ,     but  ^0^  =  ~,  very  nearly.    .     (40) 

The  angle—   between  AO  and  the  radius  vector  Od  may  be 

called  the  polar  angle. 
Again, 

-  =  tan  AOd  =  f  tan  -faO  with  great  accuracy.    .     (41) 
Example.—  Let  0  =  6°  48';  then  £0  ~  V  50', 


232  FIELD-MANUAL   FOR   ENGINEERS, 

The  formula  gives 

tan  AOd  ^-.  f  tan  1°  59'  -  .0395766. 
/.     AOd  =  2°  15'  59", 

which  is  but  a  single  second  less  than  — -. 

a 

The  series  (39)  would  give  precisely  the  same  result. 

Furthermore,    ——AOd  varies  as  0s,  nearly, 
o 

Hence  for  0  =  13°  36'     AOd  =  -£  -  08",  nearly. 

o 

Referring  to  (39),  we  see  that  -  increases  more  rapidly  than  0, 

and  therefore  more  rapidly  than  S-,  and  still  more  rapidly  than 
d*.  Hence  t  increases  more  rapidly  than  d*.  For  all  values  of 
£J  or  0  not  very  large,  it  is  plain,  however,  that  t  increases  nearly 
with  d3. 

In  the  cubic  parabola  whose  equation  is  xz  =  a?y,  AO  in  the 
figure  being  the  axis  of  x,  the  offsets  do  increase  us  the  cubes  of 
their  distances  from  0,  measured  along  AO,  increase. 

For  all  values  of  0,  not  large,  such  curve  fulfills  the  theoretical 
requirements  of  a  transition  curve  with  reasonable  exactness. 
As  0  increases,  however,  the  curve  departs  more  and  more  from 
the  true  transition  curve  and  becomes  more  and  more  unsuitable 
for  a  transition  curve. 

Moreover,  this  curve  has  a  minimum  radius  of  curvature  which 
cannot  be  passed  in  using  it  as  a  transition  curve,  and  which 
would  be  troublesome  to  many,  Furthermore,  the  deflection 
angles  are  fractional,  are  subject  to  no  simple  law,  and  therefore 
require  special  computation,  which  is  not  the  case,  as  we  shall 
see,  with  the  true  transition  curve. 

7.  To  find  the  point  of  the  curve  Omd  where  the  curve  is 
parallel  to  the  chord  Od. 

Let  p  be  the  point.  Since  at  0  the  curve  makes  the  angle  $0 
with  Od,  it  changes  direction  %0  between  0  and  p.  But  it  changes 
direction  an  amount  0  between  0  and  d.  Now  since  these 
changes  are  proportional  to  the  squares  of  the  distances  Op  and 
Od,  we  have 


THE   TRUE   TRANSilioiST    CUKVE. 


_ 
05'         °    ~  3' 

.-.  Op  =    VlOd  =  .577350^    or,     Op  =  %0d,  nearly. 
8.  To  find  the  tangents  dg  and  Og. 

We  have  dOg  =  £0, 

and  hence  .'.   0^  —  f  0,  very  nearly. 


o 
Also,  from  (18)  ,      dT  =  -  sin 


Sin  *°  (42, 


*    \*        sin  0       2    sin   0'  ' 

9.  To  find  the  length  of  any  radius  vector  or   chord  Od  and 
the  angles  between  these  chords. 
We  have 

angle  AOd  =  — , 
o 


and 


.      .     (43) 

Then  8  -  Od  =  S  vers  ~.      .  (44) 

o 

Hence  any  radius  vector  is  equal  to  the  arc  it  subtends  multi- 
plied by  the  cosine  of  its  polar  angle. 

Let  c  =  arc  Oa  =  arc  ab,  etc.;  these  arcs  being  sufficiently 
short  not  to  differ  sensibly  from  their  chords. 

Let  0  =  spiral  angle  of  Oa,  49  =  spiral  angle  of  Ob,  etc. 
Then 


chord  Oa  —  c  cos  — ,     or    arc  Oa  —  chord  Oa  =  c  vers  --, 
"o  o 

chord  Ob  =  2c  cos  — ,  or  arc  Ob  —  chord  Ob  =  2c  vers  ~r-,  etc. 
o  o 


234  FIELD-MANUAL  FOR  ENGINEERS. 

Example.—  "Lei  0  —  30'  and  c  =  100. 

Since  Oa  subtends  an  angle  0,  it  would  subtend  an  angle  20 
if  the  curvature  was  everywhere  the  same  as  at  a;  and  since 
Oa  =  100,  the  curvature  at  a  is  therefore  1°. 

Since  Ob  subtends  an  angle  of  46,  and  would  subtend  an  angle 
of  80  if  the  curvature  were  everywhere  the  same  as  at  b,  and 
since  Ob  =  200,  the  curvature  at  b  is 


Similarly  we  find  curvature  at  c  =  3°,  etc. 

Or,  since  at  a  the  curvature  =  i°,  at  b,  c,  etc.,  it  is  2°,  3°,  etc. 

Then  the  chord  Oa  —  100  cos    10'  =    99.99958; 
the  chord  Ob  =  200  cos    40'  =  199.986; 
the  chord  Oc  =  300  cos    90'  =  299.897; 
the  chord  Od  —  400  cos  160'  =  399.567,  etc. 

We  observe  that,  as  0,  40,  90,  etc.,  increase  as  I2,  2*,  3s,  etc., 
the  versines  of  these  angles  increase  as  I4,  24,  34,  nearly;  and 

0  40 

c  vers  —  ,  2c  vers  —  ,  etc.,  increase  as  I5,  25,  35,  etc.,  nearly. 
o  o 

We  can  also  find  the  long  chords  Ob,  Oc,  etc.,  from  Table  IV 
opposite  the  degrees  of  curvature  of  the  arcs  Ob,  Oc,  etc.,  at  the 
points  where  they  are  parallel  to  their  chords.  Thus  the  degree 
of  curvature  for  arc  Ob  is 

120'  X  .57735  =  69'.28; 

then  chord  Ob  —  199.986. 

For  arc  Oc  it  is  180'  X  .57735  =  103'.92; 

then  chord  Oc  =  299.897. 

For  arc  Od  it  is  240'  X  .57735  -  138'.  56; 

then  chord  Od  =  399.568. 

We  have 

4; 
o 


THE   TRUE   TRANSITION   CURVE.  235 

Then,  by  subtraction, 

40-0      30  90  -  49     '  56  166  -  99      70 


Hence  these  angles  between  the  chords  are  in  the  ratio  of  the 
odd  numbers  1,  3,  5,  7,  etc. 

10.  To  find  any  deflection  angle,  as  dbl,  at  any  point  b\  also  any 
chord  bd. 

U  being  the  prolongation  of  Ob,  we  have 

sin  bOd 
tan  dbl  =  - 


COS  bOd r-r 

dO 

Then  Odb  =  dbl  —  bOd  and  bd  may  be  found  by  the  sine  propor- 
tion. 

Or  as  follows  :  Let  c  and  Ci  represent  the  chords  Ob  and  bd,  and 
8  and  Si  the  corresponding  arcs.  Let  Oa  =  1,  and  0  —  the  spiral 
angle  for  Oa.  Then 


AOd  - 
and  hence  bOd  —  8i(28  -\-  8,)-^- (45) 


Now          sin  &d0  =  ^    sin  &tfd  =  —  sin  &(2S  -f  8^—. 
bd  c\  o 


c,         S 
Now  —  =  —  ,  nearly. 


...     bdO  =     -.  8^28  +  &)      =  /8(25  +  &),  nearly.    (46) 

Oi  o 

This  amounts  to  assuming  that  the  angles  bOd  and  bdO  are  in  the 
ratio  of  the  arcs  bd  and  bO,  which  is  very  nearly  true. 


Now  dbl  =  bOd  +  &eZ0  =  (5  +  S,)(2£  +  -Sr1)--.    .     .     (47) 

o 


236  FIELD-MANUAL   FOR 

We  may  find  any  chord,  as  bd,  with  sufficient  accuracy  from 
Table  IV,  opposite  the  degree  of  curvature  of  the  middle  of  the 
arc  bd. 

11.  To  find  the  exsec  dV,  also  TV,  etc. 

dV  =  dTsec  TdV  =  ^  sin  |0  sec  0, 
or  d  V  =  EV  -  Ed  -  R  sec  0  -  K'- 

TV  -  dTt&n  0  -  f-  sin  %0  tan  0; 

OV  =  01  -f  TV. 

12.  To  find  the  radius  of  curvature  at  any  point  of  the  curve 
Omd. 

We  have  K  =        =       ........     (48: 


Since  H'  is  a  radius  of  the  transition  curve  expressed  in  terms 
of  the  arc  Omd  and  the  angle  dgT,  the  relation  is  perfectly 
general. 

Eliminating  0  between  (1)  and  (7)  and  omitting  the  accent,  we 
have 


(49) 


This  shows  that  the  degree  of  curvature  varies  directly  with  the 
length  of  arc  measured  from  0,  which  agrees  with  the  definition 
of  the  curve. 

13.  To  lay  out  the  curve  by  ordinates,  or  offsets  from  the  tan- 
gent AO. 

We  have,  by  (18), 

ap  =  \0a  sin  2AOa; 
A'b  =  \0b  sin  2A Ob\ 
cC  -  ^Ocs'm  2AOc,  etc. 


THE   TRUE   TRANSITION    CURVE.  Zot 

Let  the  spiral  angle  of  Oa  =  30'.     Then  AOa  =  10',  AOb  =  40', 
AOc  =  90',  etc.     Hence 

ap  =    50  sin  20'       =    .291; 
bA'  =  100  sin  1°  20'  =  2.33; 

cC  —  150  sin  3°  00'  =  7.85,  etc. 

Now  set  a  100  feet  from  0  and  .29  of  a  foot  from  A0\ 
b  100  feet  from  a  and  2.33  feet  from  AO; 
c  100  feet  from  b  and  7.85  feet  from  AO,  etc. 
We  also  have,  from  (6), 

Op  =  Oa  cos  ±AOa; 
OA'  =  Obcos±AOb,  etc. 
But  these  quantities  are  not  needed. 

SPECIAL  PROBLEMS  AND  EXAMPLES. 
The  reader  will  observe  that  since  Kd  —  — - — ,  when  either  Kd 

re 

or  Omd  is  known  the  other  is  known  also. 

Problem  1.—  Given  the  length  and  degree  of  Kd,  Fig.  130,  to  find 
the  offset  AK &nd  tangent  distance  AO,  and  to  lay  out  the  curve. 

Example  1.— Let  Kd  be  a  9°  36'  curve  100  feet  long. 

. '.   Omd  =  200  feet. 
Eq.  (16)  gives 


.-.  AO  =  tL  -  .088  =  99.912, 
Eq.  (27)  gives 


=  2.789. 


. 

Oi  O  O 

Or  AST  and  J.O  may  be  taken  directly  from  Tables  XV  and  XVI. 


238  FIELD-MANUAL   FOR   ENGINEERS. 

To  LAY  OUT  THE  CURVE. 

Use  four  50-feet  chords,  for  example.    dOg  —  —  =  3°  12'  =  192'. 

o 

Set  the  transit  at  0  and  turn  off  from  AO. 

—  =  12'  for  station  (a); 
16 

12'  x  4  =  48'  for  station  (5); 
12'  X  9  =  1°  48'  for  station  (c); 

and  12'  X  16  =  3°  12'  =  ^    for  station  (d). 

o 

Or,  having  computed  the  first  deflection  angle  (12'),  find  it  in 
column  1  of  Table  XVII,  and  opposite  it,  in  columns  2,  3,  etc.,  are 
the  deflection  angles  for  stations  2,  3,  etc. 

Proceed  in  precisely  the  same  way  in  any  case.     That  is,  divide 

dOg  —  —  by  the  square  of  the  number  of  equal  chords  to  be  used, 
o 

vfhich  gives  the  deflection  angle  for  station  1;  then  multiply  this 
angle  by  22  =  4,  32  =  9,  etc. ,  for  succeeding  stations.  Or  take 
the  angles,  after  the  first,  from  Table  XVII. 

Measure  the  chords  Oa,  ab,  etc.,  as  for  a  uniform  curve. 

Example  2. — Let  the  degree  of  Kd  be  D'  =  8°  20',  and  8  =  300 
feet.  Then 

OAA  -J  Kf) 

Kd  =  ~  =  150  feet,     and    dEK  =  J^  X  8°  20'  =  12°  30'  =  0. 

A  100 

5.  =  6°  15',     ^  =  4°  10'  =  250',    and    ~  =  3°  7f. 
0.9  4 

Compute  as  above,  or  Table  XV  gives 

AK  =  5.44, 

and  Table  XVI  gives  AO  =  149.78. 

2^0 
Using  chords  of  50  feet,  the  first  deflection  angle  =  —  =  06'. 95 

or  07',  and  the  succeeding  angles  to  the  nearest  minute  are  28', 
1°  03',  1°  51',  2°  54',  and  4°  10'. 


THE    TRUE    TRANSITION    CURVED  239 

It  is  not  necessary  to  know  ^lA'iu  order  to  lay  out  the  curve;  but 
AO  must  be  known  to  give  the  beginning  of  the  curve. 

Problem  2. — Given  the  degree  of  the  offset  curve  Kd  and  the 
offset  AK,  to  find  the  length  of  the  transition  curve,  etc. 

Example.— Let  radius  EK  =  R'  =  698.73,  deg.  L>'  =  8°  12'--=  492'. 
Offset  AK  —  4  feet. 

From  (37), 

4AK  16 


3#'         2096.19 
.-.  0  =10°  37*',     ^  =  3°  32V  =  212y,    and     ~  =  2°  89$'. 

o  4 

Then 

Kd-^8=  100—,  =  100  .  ---—f  =  100 ^-^  =  129.60. 

Table  XVI  gives  d  =  .14,  and  therefore  AO  —  129.46. 

Use  five  chords,  each  •"'•'.. =  51.8  feet  in  length. 

5 

212— 

The  first  deflection  angle  is  --^-  =  8.5,  and  the  successive  an- 
gles are  34',  1°  16£',  2°  16',  and  3°  32£'. 

In  case  it  is  desired  to  use  an  offset  of  an  approximate  given 
length,  it  is  generally  sufficient  and  best  to  take  Shi  round  num- 
bers so  as  to  correspond  with  the  desired  offset  sufficiently  near, 
and  then  proceed  as  in  Problem  1.  Table  XV  very  much  facili- 
tates this  operation.  Thus  for  D'  —  8°  12'  and  S  =  250,  we  find 
(by  interpolating  for  the  .02'  above  8°  10')  offset  =  3.72  feet. 

Also, 

0  =  —  D'  -  10°  15',     ~  =  205',     and    •-  =  2°  34'. 
Then  Table  XVI  give's  d  =  .12,  and  therefore 
AO  =  ~  -  d  =  124.88. 

Problem  3. — Given  the  degree  (D')  of  Kd  and  the  tangent  dis- 
tance AO,  to  find  the  length  of  the  transition  curve  Omd,  and  the 
offset  AK. 


'MO  FtKLD-MAXUAL    FOE    ENGINEERS. 

Example.—  Let  D'=9°  36',    .'.  #'=596.83,  andlet  ^0  =  99.912. 
From  (11), 

sin  f  0  =  99.912  -5-  1342.87  =  .07440; 

.-.     0  =  9°  36'. 
From  (10), 

~  =  99.912  -^  .99912  =  100; 

.-.     £=200. 
Now  (27)  or  Table  XV  gives 

AK=  2.789. 

c  SO 

Remark.  —  If  $(or  Kd  =  -)  and  ^10  =  —  cos  —  are  given,  we 

have     cos?-  =  AO  +  ~,  which  gives  0.     Then  D'  =  ~. 
4  -c  Art 

cr 

Or,  under  8  in  Table  XVI,  find  d  =  —  —  ^10,  opposite  which,  in 
column  1,  is  7)'.     Then 


Then  ^4^  =  ^  sin  ^-;   or,  with  D'  and  S,  Table  XV  gives  AK. 

___,,.   T_/O,0  .         0  A  -rr  $  s\ 

If  5  and  ^4  A  =  —  sin  —   are  given,    sm  —  =  AK  -.  --  .     Or 
o  2  26 

under  S  in  Table  XV  find  AK,  opposite  which,  in  column  1,  is  D'  '. 
Then  as  before. 

If  AO  and  AK  are  given,  (31)  gives 


S  0 

Then  (10)  gives  —  =  AO  sec  —  .     Then  as  above. 


THE    TRUE    TRANSITION    CURVE.  241 

Problem  4. — Given  tbe  degree  D  of  the  main  curve  Af,  and  tbe 
length  of  Kd  or  of  Omd  to  find  the  offset  AK,  the  tangent  dis- 
tance AO,  etc. 

Example  I.-Let  ^#-#=599.62,  ,',  D  =  9°  33*'.  Let 
5  =  200. 

Then  J5Td  =  100,  Af>  100,  and  0  =  degree  of  Kd  =  D'  >  D. 
Assume  0  =  D  =  9°  33*'.  Then  |  =  4°  46f , 

4.°  4fia/ 
and,  from  (27),       ^^  =  100  sin  -  =  2.776. 

o 

Then  R'  =  R  -  t  =  596.844. 

.-.     D'  =  9°  36'  =  0,;     ~  =  4°  48';       ^    =  192',  etc. 
Now  AO  =  100  cos  2°  34'  =  99,912. 

Or  these  values  of  AK  and  AO  may  be  taken  from  the  tables. 

The  above  value  of  offset  is  only  approximate  ;  but  AO.  and 
not  AK.  is  used  in  laying  out  the  curve.  To  show  the  trifling 
effect  upon  Kd  and  AO  of  an  error  in  the  offset  we  now  compute 
corrected  offset. 

We  have  AK  =  t'  =  100  sin  *li®'  =  2.789. 

o 

Let  R"  and.  D"  represent  the  new  values  of  R'  and  jy. 
Then      R"  =  R  —  t'  =  596.831,     and    D"  =  9°  36'  =  IT. 
Or  Table  XV  gives  opposite  9°  33*'  t  =  2.776. 

Then  R'  =  R  -  2.776  =  596.844,    and   D'  =  9°  36'. 
Now  opposite  9°  36'  we  find  t'  =  2.789,  etc. 

It  is  seen  that  t',  the  corrected  value  of  t,  exceeds  t  by  only  .013, 
and  therefore  R",  the  corrected  value  of  R' ,  falls  short  of  R'  by 
the  same  .amount;  so  that  D"  exceeds  D'  by  only  £  of  a  second  and 
does  not  appear  in  the  result.  This  f  of  a  second  in  I)'  or  0  de- 
creases AO  by  less  than  two  units  of  the  fifth  decimal  place. 


242  FIELD-MAXUAL    FOR    ENGINEERS. 

We  thus  see  that  the  assumption,/*??'  the  purpose  of  computing 
the  offset,  that  the  degree  of  Kd  is  the  same  as  that  of  Af,  leads  to 
only  a  small  error  in  the  offset  and  to  an  exceedingly  small  error 
in  the  essential  quantities  involved;  so  that  it  is  useless  to  recom- 


pute the  offset,  etc. 
We  also  have,  from  (32), 


R-2t' 


Example  2.—  Let  AE  =  R=  67407,  .-.  D  =  8°  30'. 
Let  S  =  300,  then  Kd  =  150.     Af  >  150,  and  0  >  |Z>. 

Assume        0  =  \D  =  12°  45',     and     ^-  =  6°  22f  . 


S      ,     0         100  sin  6°  224' 

Then     AK  =  t  =  —  sm  —  =  -  -  —      -  =  5.55. 
o  /&  & 

Hence  R'  =  674.07  -  5.55  =  668.52. 


.-.  D'  =  8°  34';     0  =  \D'  =  12°  51';          =  6°  254'; 

a 

?-  =  257',     and     ~  =  3°  12f. 
o  4 


It  is  useless,  as  already  explained,  to  compute  a  corrected  offset 
t' ,  since  it  would  not  increase  D'  but  little  over  2",  nor  decrease 
AO  by  a  unit  of  the  fourth  decimal  place. 

Now  AO  =  150  cos  3°  12f  =  149.76. 

Problem  5. — Given  the  degree  of  the  main  curve  and  the  length 
of  the  part  Af  replaced,  to  find  the  offset  AK,  the  tangent  dis- 
tance AO,  etc. 

Example.— Let  AE  —  R  —  599.62,  .'.  D  -  9°  33i'.  Let  Af  = 
100. 


Then 


0  =  D ;  f  0  =  6°  22'|;  -  =  191',  and     ~  =  2 


THE   TRUE   TRANSITION   CURVE. 


243 


By  (38), 

Hence 

and 
Now 


R  vers  £0        599.63  X  .0061746 


|  +  vers 


1.3395 


=  2.764. 


EK=R'  =  599.62  -  2.764  =  596.87, 
D'  —  9°  36'. 

?-=  Ed=  100^-,  =  99.54. 
»  JJ 


Table  XVI  gives  for  D'  =  9°36'  and  8  =  200, 
|^-40  =  .09; 
.-.  AO  =  99.45. 

Use  four  chords  each  — - —  =  49.77  feet  long. 

The  deflection  angle  for  the  first  station  is  JT9^  =  12',  nearly, 
and  others  are  48X,  10?V,  and  191'. 

Problem  6. — To  replace  each  half  of  a  simple  curve  AfA'y 
Fig.  131,  of  radius  R  and  degree  D  by  a  transition  curve. 


Example.—  Let 
etc. 


=  716.2,  .-.  D  =  8°,  AEV~  %V  =  0  =  19e 


246  FIELD-MANUAL    FOR   ENGINEERS. 

Then,  see  eqs.  (45),  (46),  (47), 


ceO  =  S(2S+  Si}~\ 

o 


and  eel  =  (S  +  8^(28 


. 

o 


Now  the  transit  being  at  c,  sight  to  0,  reverse  on  I,  and  turn  off 
successive  angles  found  by  putting  8,  =1,  Si  =2,  etc.,  in  the  last 
equation.  These  angles  are 

(8  +  1)(28  +  I)6-,        (S  +  2)(2S  +  2)L  etc. 
o  o 

Then  measure  cd,  de,  etc.,  as  usual. 

The  curve  may  be  easily  retraced  from  e  toward  0. 


We  have  id  -  OcK'  =  2AOc  =  28  V- 

o 


.  (50) 
o 


w        l  >)-.     .     (51) 

Let  8'  =  8+8,.     Then,  in  (51), 

for  the  first  station  from  e  put   8  =  #'—  1,     and  Si  =  1; 

for  the  second  station  from  e  put  8  =  8'  —  2,     and  St  —  2,  etc., 

giving  the  angles      (8'  -  i)9,     (ff  —  f  )20,    etc. 

Turn  off  these  angles  in  succession  from  the  tangent  et,  and 
locate  d,  c,  etc.;  otherwise  as  usual. 

Problem  9.  —  To  substitute  a  transition  curve  for  an  end  portion 
of  a  main  curve  without  moving  the  rest  of  the  curve. 


Again, 

cet  =  Oet  —  Oec  =  2AOe  -  ceO 


THE  TRUE  TRANSITION  CURVE. 


247 


Let  Aide  (Fig.  133)  be  a  part  of  the  main  curve  tangent  to  0 T 
A.i.  Suppose  the  curve 


deK, 

--  R,  to  be  run  from 
some  points  of  tliecurve 
Aide,  through  an  angle 
KE'd-AiEd  =  0,,  and 
draw  dHHi  parallel  to 
OT. 

Suppose  a  transition 
curve  emO,  having  a 
central  angle  KE'e  —  0, 
to  connect  this  curve 
with  the  tangent  OT  at  FlG-  133- 

0.  The  point  d  must  be  taken  far  enough  from  Ai  to  give  room 
for  the  transition  curve  between  d  and  the  tangent  AO;  and  ft' 
cannot  be  <  %R,  for,  if  so,  KH  would  be  <  %AH,  or  AK  would 
be  >  ±AH,  and  no  transition  curve  could  connect  Ked  and  the 
tangent  AO. 

Draw  en  parallel  to  OT.     Then 

Kn  —  \An. 
In  general  we  have 

AH  —  AiH!  =  R  vers  0,,     HK  =  R'  vers  0,. 


KH 
•  AH 


(52) 


If  R'  =  f  R,  KH  =  %AH&ud  the  transition  curve  begins  at  d. 
If  R'  >  |/?,  KH  >  \AH>  or  AK  <  \AH,    and    the    transition 
curve  begins  at  some  point  between  JTand  d. 
We  have,  from  the  above, 

AK  =  AH  -  KH  =(R-  R')  vers  0,. 
Also,  nK  =  SAK  =  3(R  -  R')  vers  0t. 

HK  R' 

"  nK    ~  3(7?  -  R'f 


246  FIELD-MANUAL    FOR   ENGINEERS. 

Then,  see  eqs.  (45),  (46),  (47), 
cOe  =?& 

ceO  =  82 


and  eel  =  (8+  8i)(28  +&),y. 

o 

Now  the  transit  being  at  c,  sight  to  0,  reverse  on  I,  and  turn  off 
successive  angles  found  by  putting  81  =  1,  Si  —  2,  etc.,  in  the  last 
equation.  These  angles  are 


(8  +  1}(28  +  1)-.         (8  +  2)(2S  -f  2)-,  etc. 

o  o 

Then  measure  cd,  de,  etc.,  as  usual. 

The  curve  may  be  easily  retraced  from  e  toward  0. 

We  have  td  =  OcE'  =  2AOc  =  28^~. 

o 


.  (50) 
Again, 

Get  =  Oet  -  Oec  =  2AOe  -  ceO 


-  -  8(28+  St)-  =  8^8+  28l.     .     (51) 

Let  8'  =  8  +  8..     Then,  in  (51), 

for  the  first  station  from  e  put  £  =  Sr  —  1,     and  &  =  1; 

for  the  second  station  from  e  put  8  =  8'  —  2,     and  Si  =  2,  etc., 

giving  the  angles      (8f  -  i)0,     (£'  -  $)29,    etc. 

Turn  off  these  angles  in  succession  from  the  tangent  et,  and 
locate  d,  c,  etc.;  otherwise  as  usual. 

Problem  9,  —  To  substitute  a  transition  curve  for  an  end  portion 
of  a  main  curve  without  moving  the  rest  of  the  curve. 


THE   TRUE   TRANSITION   CURVE. 


247 


Let  Aide  (Fig.  133)  be  a  part  of  the  main  curve  tangent  to  OT 
&tAi.  Suppose  the  curve 
deK,  with  d£J'=R'<dE 
--  R,  to  be  run  from 
some  point^  of  thecurve 
Aide,  through  an  angle 
KE'd= AiEd=Oi>  and 
draw  dHHi  parallel  to 
OT. 

Suppose  a  transition 
curve  emO,  having  a 
central  angle  KE'e  =  0, 
to  connect  this  curve 
with  the  tangent  OT  at 
0.  The  point  d  must  be  taken  far  enough  from  Ai  to  give  room 
for  the  transition  curve  between  d  and  the  tangent  AO;  and  R' 
cannot  be  <  $R,  for,  if  so,  KH  would  be  <  $AH,  or  AK  would 
be  >  \AHt  and  no  transition  curve  could  connect  Ked  and  the 
tangent  AO. 

Draw  en  parallel  to  OT.     Then 


FIG.  133. 


In  general  we  have 


AH  - 


Kn  =  \An. 

Tl=  R  vers  0,,     HK  =  R'  vers  0,. 
KH 


•  AH 


K 
R' 


(52) 


If  R'  =  f  R,  KH  =  ^AH&nd  the  transition  curve  begins  at  d. 
If  R'  >  £#,  KH  >  \AH>  or  AK  <  ±AH,    and    the    transition 
curve  begins  at  some  point  between  .ZTand  d. 
We  have,  from  the  above, 

AK  =  AH  -  KH  =  (R  -  R')  vers  0,. 
Also,  nK  =  3AK  =  3(72  -  R')  vers  0^ 

HK  R' 


nK 


B(R 


248  FIELD-MANUAL  FOR   ENGINEERS. 


But 


IIK       Rf  vers  0,       vers  0 


nK        R'  vers  0        vers  0  ' 

vers  Oi  _  I? 

vers  0   -•'-  3(7?  —  .#')' 

or  vers  Ot  =  ^         vers  0  —  — — —  vers  0.  (53) 

Since  .fiTcZ  and  Aid  subtend  the  same  angle, 

Kd  _  R  __  D 
Aid  =  R  ~!y' 

We  have  seen  that  if  from  any  point  d  of  a  curve  Aide  of  radius 
R  a  curve  deK  be  described  with  radius  R'  —  $R,  then  a  transi- 
tion curve  connecting  Aide  with  the  tangent  AOi  will  begin  at  d. 

When,  however,  A\de  is  a  sharp  curve,  the  change  from  radius 
R  to  radius  |J?  would  be  rather  too  sudden,  and  therefore  it  is 
best  to  take  R'  >  ^R  and  commence  the  curve  at  a  corresponding 
point  6  given  by  eq.  (53). 

R'  =  *R,     or     R'  =  IR,     or     R'  =  -f^R 
are  practical  values  for  R'. 

7-7  7f  TJt 

The  ratio  —^  =  —^ =7    shows   that    the    larger  R'   is    the 

nJ\.        o(.K  —  -K  ) 

J-TTT 

larger  —=  is,  and  therefore  the  longer  the  connecting  branch  de. 

It  is  not  necessary  to  run  the  curve  deK  much  beyond  e;  but  if 
run  to  jfif,  we  thus  determine  A,  which  is  opposite  to  K.  In  this 
case  the  position  of  T  is  not  needed. 

A  short  solution  of  this  problem  sufficiently  accurate,  unless  0 
is  very  large,  is  the  following  : 

We  have  approximately 


nK'     "'     """  ~  •"M.,,E>-  /  •  •     • 


THE  TtiuE  TRAHsmoN  CURVE.  ^49 

This  formula  gives  only  slightly  too  small  a  value  for  Kd. 

Example  1.—  Let  R  =  954.9  (D  —  6°),  and  Ome  =  200. 

Let  Rf  =  Ty?,  then  D'  =  ^D  =  6°  40'.     To  find  d,  e,  and  0, 
and  to  locate  the  curves. 

We  have       KE'e  =  0  =  D'        =  D'  =  6°  40'; 


7?' 
vers  Ol  =  57^  -  —  vers  0  =  3  vers  0  =  .02028; 


.-.     Ol  =  11°  56'. 
Hence  A,d  =  100^-^  =  192.7; 

D 

Kd  =  ^A,d  =  &Aid  =  173.4; 

de  =  Kd-  Ke  =  73.4. 
Or,  by  (54),  Kd  =  100  \ft  =  173.2; 

de-Kd~Ke~  73.2; 
Aid  =  V  X  173.2  =  192.4,  etc. 

Now          AO  =  -  cos  —  t=  100  cos  1°  40'  =  99.96, 
or  TO  =  S  cos  |0  =  200  cos  3°  18'  =  199.700 


Also,  -  s=  133'. 

o 


For  four  equal  chords  the  first  deflection  angle  is 


Other  deflection  angles  are  found  as  heretofore, 


FIELD-MANUAL   FOR   ENGINEERS. 

Measure  off  AO  =  99.96  feet  or  TO  =  199.70  feet ;  set  the  tran- 
sit at  0.  and  locate  the  transition  curve  Ome  as  usual. 

Then  observing  that  Oeg'  —  f  0,  giving  the  position  of  the  tan- 
gent at  e,  locate  the  branch  de. 

Example  2.— Given  R  =  716.2  (D  =  8°),  R'  =  ftf,  .-.  D'  = 
|Z>  =  9°,  and  S  =  300  feet,  to  find  the  tangent  points  d,  e  and  0, 
and  to  locate  the  curves. 

o 

Since  Ke  =  r-  =  150  feet, 

B 

KE'e  =  \V  =  13°  30'  =  0. 

vers  Oi  =  0773 JJT  vers  ^  =  I  vers  ^  =  '07368. 

...     Ox  =  22°  08'  =  22°.13. 

Then  Kd  =  100—^-  =  245.9, 


and         de  =  Kd-  Ke  =  95.9  ft.  ;  J^  =  %Kd  =  276.6. 

Or,  first  find  A^d,  then  JT(Z  and  de  as  in  the  preceding  example. 

8         0       300  cos  3°  22' 
Now        ^0  =     -cos-  = =149.74. 


Also,  -  =  27<X. 

o 

270' 
For  six  equal  chords  the  first  deflection  angle  =  —  =  7'.  5. 

Locate  the  transition  curve  Ome,  and  then  the  branch  de  as  shown 
above. 

If  KE'e  =  0  is  given  instead  of  8,  we  have 


which  gives  8.     Then  find  Oi  as  before,  etc. 


THE   TRUE   TRANSITION    CtTRVti.  5 

If  it  is  desired  to  make  a  given  offset  t,  we  may  find  S  by  Table 
XV,  corresponding  to  t  and  D' ,  and  then  proceed  as  before. 

COMPARISON  BETWEEN  THE  TRUE  TRANSITION  CURVE  AND  THE 
COMPOUND  CURVE  COMMONLY  USED  FOR  A  TRANSITION  CURVE. 

This  compound  curve  is  one  of  any  number  of  branches,  the 
curve  being  "compounded  at  the  end  of  each  chord." 

The  degree  of  the  first  branch  is  made  the  common  difference  of 
the  degrees  of  the  curve  of  the  succeeding  arcs. 

Thus  suppose  the  chords  each  100  feet  long,  and  the  degree  of 
the  first  branch  10'.  Then  the  degrees  of  the  succeeding  arcs  are 
20',  30',  etc. 

Such  a  curve  inclines  to  the  tangent,  at  all  points,  too  much, 
the  inclination  at  the  middle  of  any  chord  being  precisely  what  it 
should  be  at  the  end  of  the  chord. 

This  has  the  effect  of  making  the  tangent  distance  somewhat 
too  short  and  the  offset  much  too  long. 

Thus  at  the  end  of  the  tenth  chord  the  approximate  results  are  : 
tangent  distance  —  997.32  and  offset  =  55.89.  Whereas  the  true 
tangent  distance  =  998.82  and  offset  =  48.40. 

The  errors  increase  with  the  sharpness  of  the  curve  and  with 
its  length. 

It  is  to  be  noted  that  the  true  transition  curve  is  continuous  ; 
whereas  the  approximate  transition  curves,  used  as  such,  are  dis- 
continuous. Any  point  on  the  former  can  be  determined  at  once 
and  without  reference  to  other  points  ;  whereas,  in  the  latter, 
points  on  the  curve  must  be  found  consecutively.  To  find  a  point 
at  the  extremity  of  a  chord,  for  example,  points  at  the  extremities 
of  all  preceding  chords  must  be  found. 

Thus  the  laying  out  of  such  a  curve  is,  for  this  among  other 
reasons  referred  to.  necessarily  laborious. 

Mr.  E.  W.  Crellin,  C.E.,  now  of  Des  Moines,  Iowa,  explained 
the  nature  and  to  some  extent  the  properties  of  the  true  transition 
curve  in  an  article  in  the  Transit,  published  April,  1890,  by  the 
State  University  of  Iowa. 

The  author  is  not  aware,  however,  that  any  general  discussion 
of  the  curve  has  heretofore  been  made ;  that  its  general  proper- 
ties have  been  found  ;  that  practical  formulas  for  it  have  been 
deduced  ;  and  that  its  applicability  has  been  shown. 

A  writer  on  transition  curves  says  that  "the  theoretic  curve 


252  FlELD-MAKUAL   FOft   ENGINEERS. 

Avhich  should  be  adopted  is  a  form  of  the  elastic  curve,  Avhich,  oil 
account  of  the  trouble  in  locating  it,  has  been  supplanted  by 
various  approximations,  such  as  the  curve  of  sines,  parabolas, 
etc.;  these  being  easier  to  locate  in  the  field." 

The  true  transition  curve,  as  we  have  seen,  is,  on  the  contrary, 
much  more  easily  located  than  any  of  the  approximate  curves 
used. 


CHAPTER  X. 
CALCULATION  OF  EARTHWORK. 

THIS  chapter,  for  obvious  reasons,  does  not  constitute  a  com- 
plete treatise  on  earthwork. 

The  object  here  sought  is  to  treat  in  the  most  simple  manner, 
and  by  means  of  the  shortest  formulas,  the  usual  cases  that  arise 
in  practice. 

To  compute  the  contents  of  an   excavation  it  is,  in  general, 


FIG.  134. 

necessary  to  divide  it  into  a  number  of  prismoids  by  cross-sections 
at  the  stations  and  at  other  suitable  points. 

A  prismoid  is  a  volume  the  ends  or  bases  of  which  are  parallel 
planes,  and  the  lateral  surfaces  are  either  planes  or  warped  sur- 

253 


254 


FIELD-MANUAL   FOR   ENGINEERS. 


faces,  such  as  may  be  generated  by  a  right  line  moving  along  two 
lateral  edges  as  directrices,  these  edges  being  right  lines  joining 
corresponding  angles  or  corners  in  the  bases. 

Thus  let  Fig.  134  represent  a  prisnioidal  excavation  between 
the  cross-sections  at  C  and  C". 

HabKC  represents  one  end,  and  the  same  letters  accented  repre- 
sent the  other  end.  ciba'b'  represents  the  roadbed,  aa'HH'  and 
WKK'  represent  the  side  slopes.  CC'HH'  represents  the  surface 
of  the  ground  on  the  left  of  the  center  line  CC',  and  GG'KK'  the 
same  on  the  right. 


FORMULAS  FOR  SECTIONAL  AREAS. 

Let  b  =  the  width  of  roadbed  ; 
S  —  slope  of  sides  ; 
c  =  depth  at  center  stake  ; 
7i  and  k  =  depth  at  side  stakes  ; 
d  and  e  =  distances  from  center  to  side  stakes. 

A.  For  ground  level  laterally  the  section  abllK  (see  Fig.  135) 
is  a  trapezoid,  having  an  average  width  of 


Ad  -f  oh  =  cS  -f-  6,     and     area  =  a  =  c(cS 


(1) 


E 

FIG.  135. 

B.  When  the  ground  slopes  are  not  the  same  on  opposite  sides 
of  the  center  line  as  shown  in  Fig.  136,  we  find  as  follows  : 

Prolong  the  side  slopes  Ha  and  Kb  to  intersect  in  E. 

^=I  =  JL 


CALCULATION    OF    EARTHWORK. 


Then     area  GHEK  =  \GE  X  AB  =  \{e  +  £S(d  + 


255 


area  of  abE  =  aD  X  DE  = 


6* 
48  ' 


Then     area  CEabk  =  a  =  ~(c  +  -L\d  +  «)  -  -|1. 

We  observe  that  —  ^  and  —  —  are  constant  for  the  same  slope 
2a  4o 

and  width  of  roadbed.     Represent  the  former  by  i  and  the  latter 
by  a'  and  we  have 

area  abHK  =  a  =  \(c  +i,}(d-}-  e)  -a'.       .     .  (2) 
For  another  section 

ai  =  i(ci  +  0(di  +  «,)  -  a'.       ...  (3) 

Let  GE  —  c-\-  i~  C,    and   ^i#  =  ^  +  e  —  D,  and  we  have 

area  of  CHabK  =  a  =  |(7Z>  -  a'  .......  (4) 

Similarly  for  another  section 

a,  =  itf,D,  -  a'.     •  .......  (5) 


These  are  the  simplest  formulas  possible  for  this  case,  since 
each  requires  but  a  single  multiplication,  while  any  other  formula 


FIELD-MANUAL   FOR   ENGINEERS. 


requires  two  multiplications.     These  formulas  have  been  used  by 

the  author  for  more  than  thirty  years.     See  correspondence    in 

the  Engineering  News,  vol.  32,  page  73. 

When  the  surface  of  the  ground  is  irregular  between  the  center 

and  side  stakes  it  is  best  to  divide  the  cross- section  into  trapezoids 

by  dropping  perpendiculars 
to  the  roadbed  at  those 
stakes,  and  at  each  break  in 
the  surface  between  them, 
thus  forming  a  number  of 
trapezoids  as  shown  in  Fig. 
137.  Then  the  area  of  the 
section  is  equal  to  the  area 


F       b       H 
FIG.  137. 
of  the  trapezoids  less  the  area  of  the  triangle  bBK. 


FORMULAS  FOR  REGULAR  EXCAVATIONS  AND  EMBANKMENTS. 

A.  The  volume  of  a  prismoid  may  be  exactly  calculated  by  the 
prismoidal  formula,*  which  is 


=     -(«  -f  4m 


(6) 


in  which  V=  volume,  I  =  length  ;  at  a,  =  the  areas  at  the  ends, 
and  m  =  the  area  of  the  middle  section. 

To  find  the  volume  in  cubic  yards,  divide  the  above  by  27. 

Since  the  linear  dimensions  of  the  middle  section  are  arith- 
metical means  of  the  corresponding  dimensions  of  the  end 
sections,  we  have 


in  which  a!  =  the  area  of  the  triangle  under  the  roadbed. 

*  The  late  Prof.  W.  M.  Gillespie  proved,  many  years  ago,  that  the  pris- 
moidal formula  is  applicable  to  a  prismoid  having  one  warped  surface  ; 
and  the  late  Prof.  De  Volson  Wood  afterwards  proved  that  the  same 
formula  is  applicable  to  a  prismoid  having  two  warped  surfaces. 

The  author  proved  (in  1869)  that  the  formula  in  question  applies  directly 
to  a  prismoid  having  any  number  of  lateral  surfaces,  any  number  or  all  of 
which  are  warped. 

Mr.  A.  M.  Bannister,  C.E.,  of  Lansing,  Mich.,  Mr.  Cornelius  Donovan,  C.E., 
of  Port  Eads,  La.,  and  others  are  acquainted  with  the  facts.  An  elegant 
proof  was  since  given  by  Prof.  George  Bruce  Halsted  of  Austin,  Texas. 


CALCULATION   OF   EARTHWORK.  257 

B.  Substituting  the  values  of  a,  a\ ,  and  in  in  (6),  we  find 

vol  =  j(JL  +  — )  -  ~(C  -  C&D  -  A);      .    (8) 

vol  =  lm  +g|(0-  C,}(D  -  A) (9) 

We  observe  that  C  —  Ci  =  c  —  Ci, 

—  -| -j    is    called  the    ''end  area  volume,"  and  lin  the 

2  / 

middle  area  volume.  The  above  equations  show  that  the  former 
differs  from  the  true  volume  twice  as  much  as  the  latter  and  in 
the  opposite  sense.  This  may  be  shown  symbolically  as  follows  : 

—(a  +  4m  -  a,)  — ^(«  +  «0  =  gr(2w  -  a  -  ^); 

—(a  +  4m  +  oi)  —  J/w.  =  -^-(a  -f  «i  -  2w). 
6  b 

The  last  of  these  equations  is  one  half  of  the  first  with  a  contrary 
sign. 

In  case  the  prismoid  is  level  laterally  we  have,  from  Fig.  135, 

D  =  AB  =  ab  +  2Aa  =  b  +  2cS. 
Similarly  A  =  &  +  2dS. 

.:     D  —  D!  —  2S(c  —  c,). 


=  J  (|  +  f)  -  ^(e  -  «,)'. 


Hence  vol  =  I    -  +  -9-     -  — -  (c  -  c,)\   .     .     .     (10) 


70 

or  vol  =  lm+  j^(c  -  Cl)' (11) 

To  generalize  equation  (8): 

Let  a,  a\ ,  «-a  .  .  .  art  represent  areas  of  sections  ; 

c,  Ci ,  Ci    ...  cn  represent  center  heights  ; 

and       d,  d\ ,  di  .  .  .  dn  represent  tl  e  extreme    width  of  sections, 
as  AB,  Fig.  136, 


258 


FIELD-MANUAL   FOR  ENGINEERS. 


Then  expressing1  volumes  of  successive  prismoids,  and  adding, 
we  get 


vol  =  l(\a  -f  ai  + 
l   f 


n-!  -  dn)].     (12) 


The  areas  of  one  half  of  the  end  sections  appear,  and  the  whole 
areas  of  other  sections. 

The  correction  for  each  prismoid  must  be  taken  separately. 

Let  Fig.  138  represent  a  longitudinal  section  of  a  "cut" 
through  the  center  line.  Drop  the  perpendiculars  CD,  C\D\,  etc., 
and  at  the  middle  points  of  DD\,  DiD?,  etc.,  erect  perpendiculars 
i,  MJ?,  etc.,  to  meet  horizontal  lines  through  C,  (7,,  etc. 


D      M      Da     M!    Do     M2    D3    M4    D4    0 
FIG. 138. 

Then  the  first  part  of  formula  (12)  gives  the  volume  represented 
by  GDME  -f  E1MM1F  +  E^M^N^Q  -f  etc.,  and  the  second  part 
furnishes  the  required  corrections. 

Example.—  Let  b  =  20  and  8  =  1.  Then  (see  Fig.  136)  DE  = 
10  and  area  abE  —  100.  Commencing  at  C,  let  the  center  heights 
be  3.0,  6.0,  7.0,  5.0,  and  2.5,  and  distances  out  be 
26.0,  33.2,  34.8,  31.6,  and  26.0. 

Now,  by  (2)  or  (4), 

\a  =  i  X  13  X  26  -  50       =    34.5 

0,1  =  |  X  16  X  33.2  -  100  =  165.6 

aa  =  $  X  17  X  34.8  -  100  =  195.8 

a3  =  |  X  15  X  31.6  -  100  =  137.0 
$04  =  i  X  12.5  X  26  -  50    =    31.25 
Then  ~wTs  X  100  =  56410, 


CALCULATION    OF    EARTHWORK.  259 

Also,  (c  -  c,)(d  -  d.)  =  3.0  X  7.2  =  21.6 
(Cl  _  c9)(dl  -  da)  =  1.0  X  1.6  =  1.6 
(c-9  -  c3)(d*  -  da)  =  2.0  X  3.2  =  6.4 
(c,  -  c,)(ds  -  d<)  =  2.5  X  5.6  =  14.0 

43.6    X  -W-  =  363. 
Therefore  volume  =  56052  cu.  feet. 

These  quantities  may  be  taken  at  once  from  suitable  tables, 
thus  requiring  only  a  few  additions  to  give  volumes  of  about  as 
many  prismoids. 

The  prismoids  ending  at  0  and  0'  are  best  computed  separately. 

To  compute  CDO. 

Let  DO  —  I'  —  75.     The  distance  out  at  0  =  b  =  20. 

Then  \ol'  =  34.5  X  75  =  2587.5 

and  f|(c  -  0)(d  -  6)  =  6.25  X  3  X  6  =    112.5 

.',  volume  =  2475.0 

If  the  ground  is  not  level  laterally  in  the  region  of  0  (or  0'},  we 
must  take  cross-sections  where  the  center  line  or  either  edge  of 
the  roadbed  comes  to  grade  and  compute  the  volumes  between 
the  cross-sections  separately. 

Generalizing  eq.  (10),  we  have 


~      [(«  -  *»)*  +  (C»   -  <!•)'  +  -  •  •  (Cn-l  -  Cn)1]-      (13) 

The  areas  may  be  computed  by  eq.  (4),  and  the  quantities  within 
the  brackets  by  a  table  of  squares  ;  or  all  may  be  found  by  suit- 
able earthwork  tables. 

SPECIAL  FORMULAS  AND  CASES. 

Let  CDK  '  .  .  .  C",  Fig.  139,  represent  a  triangular  prismoid. 

Let  CD  =  c,  C'D'  =  Cl,  KB  =  K,  K'B'  =  K,t  DB  '=  df  and 
D'B'  =  d,. 

Let  a  and  a\  represent  the  areas  of  CKD  and  C'K'D'\  m  —  the 
area  of  the  middle  section;  I  =  the  length. 


260  FIELD-MANUAL   FOB   ENGINEERS. 

We  readily  find 


vol.  =       cd  -f  - 


0,  +  4w),    (14) 


c,       -c  - 


(14') 


(14") 


FIG.  139. 

These  equations  will  evidently  give  the  volumes  on  both  sides 
of  the  center  line  by  letting  d  and  dl  represent  the  sum  of  the 
distances  out  at  the  respective  ends  of  the  prismoid. 

The  above  equations  also  give  the  volume  of  KBD  .  .  .  K'  by 
substituting  JTand  K1  in  place  of  c  and  c'. 

Hence 


vol  KBD  ...K'  =  - 


T-f  Ki)(d  +  d,}} 


-(a  -f  a  i  -(-  4w), 


(15) 


CALCULATION    OF    EARTHWORK. 


-  Ks).     (15") 


APPLICATION  OF  GROUP  (15). 

If  d  =  rfi  and  K  —  Ki,  tlie  prismoid  becomes  a  prism  and  any 
equation  of  tlie  group  gives 

vol  =  l1^  =  la. 

& 

IKd       la 
If  rfi  =  Ki  -  0,  the  figure  is  a  pyramid  and  vol  =:  —    =  —  . 

!£--.  =  —  or  dKj  —  d^K,  tlie  prismoid  becpmes  a  frustum  of  a 
d\       KI 

id  and 

vol  =  l-(dK+  diKi  +  d,K\ 


or  ^ol  =  -(dK  +  diKi  +  did). 

The  formula  given  in  the  books  for  the  volume  of  a  frustum  of 
a  pyramid  is 

vol  =  --(a  + 
o 


rl'he  formula  above  is  much  more  simple. 

FORMULA  FOR  THE  VOLUME  OF  ANY  FRUSTUM  OF  A  PYRAMID. 

Let  d  and  K  represent  any  two  sides  of  one  base,  and  di  and  KI 
corresponding  sides  of  the  other  base. 
Let  a  =  ndK,  then 

«j  =  ndiKi  ,      and        aa^ 


jjix  VTI.LN  Hi  r- 


or  putting  (ZjJT  =  dK},  we  "have 
,     and 


Hence        (a  -f  |/oa,  31  a»)  = 

o  o 


In  this  substitute     n  =  -^7=      or     w  =  ~   *  . 
dK  d.Ki 

In  the  above  ?i  =  i,  and     ,*,  vol  z=  -(dK  -\-  dtK\  -f- 

b 


Adding  (14")  and  (15"),  we  Lave 
vol  =  ld 


or  1(16) 


""i/V  -vi     i    -i 

since  d(c  -\-  K)  =  2a,     and     di(Ci  -(-  JTi)  =  2«i. 

This  applies  to  a  prismoid  having  vertical  sides  and  a  trape- 
zoidal base. 

If  d  =  di,  the  base  is  a  rectangle,  and  we  have,  from  the  above, 

vol  =  ™(c  +  K  +  e,  +  Ki)  =  l(j  +  ~V    -          (17) 
For  n  prismoids  of  common  length  I  and  width  d  we  have 

vol  =  ?| 


,0i=^+ai  +  a,+     +«„,  +  '«„) 

If  di  —  0,  the  edges  at  C'  and  K'  coincide,  c\  =  K\,  and  we 
have,  from  (16), 


CALCULATION    OF    EARTHWORK. 


263 


The  figure  is  a  wedge,  —  being  the  area  of  the  base  PDD  ,  and 

—  the  average  height  of  the  edges. 
o 


If  c  =  K,  and  d  =  K1}  in  Fig.  li'>»,  (16)  becomes 

I  . 


vol  =  -(cd 


-&).      .     .     (20) 


We  observe  with  reference  to  all  prismoids  that  the  bases  may 
occupy  any  relative  position  in  the  parallel  planes  without  in  the 
least  affecting  their  volumes. 

LOADED  FLAT  CARS,  PILES  OF  STONE,  ETC. 

Eq.  (20)  gives  the  volume  of  a  fiat  car  loaded  with  earth,  gravel, 
etc.,  or  a  pile  of  stone,  etc.,  c  and  d  being  the  dimension  of  either 
base,  and  c\  and  d\  those  of  the  other  base. 

Usually  the  side  slopes  in  such  cases  are  nearly  uniform,  but 
the  formula  applies,  as  we  have  seen,  if  each  slope  is  uniform, 
whether  they  are  equal  or  not,  and  also  when  the  side  slopes  are 
warped  surfaces. 

The  volume  of  a  borrow-pit  divided  into  prisms  having  equal  or 
equivalent  bases,  of  area  =  A,  is  readily  found  by  a  single  calcu- 
lation. 

Let  Fig.  140  represent  the  base  of  such  an  excavation,  the  heights 
at  the  corners  being  denoted 
by  the  letters  there  placed. 
The  total  volume  is  equal  to 
±A  multiplied  by  the  sum  total 
of  the  sum  of  the  corner 
heights  of  the  several  prisms. 
Into  this  total  sum  the  corner 
heights  a,  cij,  b,  £3,  d,  d3  will 
enter  but  once,  being  found  in 
but  one  prism;  c,  c3,  d^  and  c?2 
will  enter  twice,  being  common 
to  two  prisms;  ?>i  and  62  will 
enter  three  times,  being  com- 
mon to  three  prisms.  Finally,  c 
being  common  to  four  prisms. 


C] 


d3 
FIG.  140. 
and  c-2  will   enter  four  times, 


264  FIELD-MANUAL   FOB,   ENGINEERS. 

Representing  these  sums  in  order  by  S)}  S^,  83,  and  $4,  we  Lave, 
for  all  the  prisms, 

vol  =  $A(Si  +  &  +  £3  +  &)  .....     (21) 


ENDS  OF  EMBANKMENTS  OR  "DUMPS." 

Let  Fig.  141  represent  an  embankment. 

Let  c  and  Ci  represent  the  center  heights  and  d  and  di  the  hori- 
zontal widths  of  HK  and  IfiKi. 


FIG.  141. 
Draw  a#i  and  bid.     Then,  by  (8), 


vol  abHCK .  .  .  Hi  GiKi  =  I-  -  —c(d  —  di 

since  Ci  =  0,     and     tii  —  0. 

ltd-  dlt 

vol  of  dump  =  I- (23) 

GROUND  IRREGULAR  LATERALLY. 

Let  Fig.  142  represent  the  part  of  a  prismoid  on  the  right  of  the 
center  line  CO'  for  the  case  supposed,  which  requires  the  nitev- 
mediate  heights  at  D,  E,  D1 ',  and  R'.  The  toiai  volmt  e  is  equal  to 
the  volume  of  the  prismoids  whose  bases  are  ntFH/i,  FGLII,  and 
GBB'L,  less  the  volume  of  the  sub-slope  prismoid  bBK . . .  b'B'K', 


CALCULATION    OF    EARTHWORK. 


265 


In  general  the  volumes  of  the  former  are  found  by  (16).  When, 
however,  the  widths  at  the  two  ends  of  the  prismoid  are  equal, 
(17)  applies. 

When  more  heights  are  taken  at  one  end  than  at  the  other,  the 
width  of  one  or  more  prismoids  at  one  end  is  0  and  (19)  applies. 


The  volume  of  the  sub-slope  prismoid  is  found  by  any  equation 
of  group  (15). 

MIXED  WORK,  EXCAVATION  AND  EMBANKMENT. 

Suppose,  for  example,  there  is  excavation  on  both  sides  and  em- 
bankment on  one  side  only. 

Let  Fig.  143  represent  the  cross-section  at  one  end  of  the  pris- 


FIG.  143. 


FIELD-MANUAL   FOR   ENGINEERS. 

moid.     CDb K  shows  the  cut  on  the  right,  CDM  that  on  the  left 
and  allM  the   fill   on  the  left.     Db  is    the   half- width  of    road- 
way in  the  cut,  and  aD  the  half- width  in  the  fill  and  cut.     11CK 
shows  the  surface  of  the  ground,  and  a H  said  bK  the  side  slopes. 

The  volume  of  the  cut  on  the  right  is  found  at  once  by  (8),  ob- 
serving that  in  this  case  a  and  «.  represent  the  areas  on  one  side 
only  of  the  center  line,  and  Z>aud  DI  tue  distances  out  on  one  side 
only. 

The  volume  of  the  cut  on  the  left  is  found  by  (14),  by  substitut- 
ing the  values  of  CD  and  DM  for  c  and  d,  and  making  similar 
substitutions  for  c\  and  di  at  the  other  end  of  the  prismoid. 

The  volume  of  the  embankment  is  found  by  means  of  the  same 
equations  by  substituting  the  values  of  aM and  All  in  place  of  c 
and  d,  etc. 

In  case  a  fill  is  on  both  sides  of  the  center  line  and  a  cut  on  one 
side  only,  the  same  formulas  of  course  apply. 

A  cross-section  must  be  taken  at  the  point  where  the  surface  of 
the  ground  intersects  the  center  of  the  roadbed.  At  this  point 
the  triangle  CDM  vanishes,  CDbK  becomes  DbK,  and  HaM  be- 
comes HaD. 

Then  the  volume  of  the  cut,  of  which  CDbK  shows  one  end  and 
DbK  may  represent  the  other,  is  given  by  (8),  in  which  Ci  =  0. 

The  volume  of  the  fill,  of  which  allM  shows  one  end  and  aHD 
may  represent  the  other,  is  found  by  (14),  as  before. 

The  volume  of  the  pyramid,  of  which  CDM  shows  one  end  and 
the  point  D  represents  the  apex,  is  given  by  an  equation  follow- 
ing group  (15),  and  is 

IKd 

vol  =  — , 

in  which  K  =  CD,     and    d  =  DM. 

The  prisrnoids  adjacent  to  the  cross-section  where  the  change 
considered  occurs,  but  on  the  opposite  side  of  it  to  those  just  con- 
sidered, are,  it  is  evident,  computed  in  the  same  way  and  by 
means  of  the  same  formulas. 

A  cross-section  must  be  taken,  also,  where  the  surface  of  the 
ground  intersects  the  side  of  the  roadbed.  In  this  case  the  same 
formulas  apply  to  the  prismoids  on  either  side  of  the  section  in 
question,  and  adjacent  to  it,  in  precisely  the  same  way  as  in  the 
case  just  considered. 


CALCULATION    OF    EARTHWORK.  267 

CORRECTION  OF  EARTHWORK  FOR  CURVATURE. 

Suppose  the  center  line  of  the  prismoid  (Fig.  139)  is  an  arc  of  a 
circle  of  radius  R. 

Let  e  =  the  length,  measured  on  the  line  CC',  of  a  very  short 
prismoid,  at  CDK  or  C'D'K',  or  at  the  middle  section. 

Then  the  lengths  measured  through  the  centers  of  gravity  of 
these  prismoids  are  as  follows  : 

The  length  at  CDK  is 
The  length  at  C'D'K'  is 
and  that  at  the  middle  section  is  e 


R 

It  is  evident,  however,  that  we  may  consider  the  lengths 
through  the  centers  of  gravity  of  the  elementary  prismoids  to  he 
the  same  as  their  lengths  along  CC'  if  we,  at  the  same  time, 
increase  the  areas  of  the  middle  and  of  the  end  sections  accord- 
ingly. 

We  must  write,  therefore,   — -t —  A  in  place  of  A, 

H 

R+Sdi 
— — AI  in  place  of  Ai , 

R  +  tfm 

and  — m  in  place  of  m. 

K 

Hence 

I  (R  +  Id  .       R  +  Wt   .        .11  +  ±dm    \ 
truevol=r_l — _ — A-\ -^ — A  +  4 ~ — mj,  .     (25) 

and 

the  excess  of  vol  =  _[_*_  j[  _j_  L-l^  -j-  4*£L-j|A      (26) 
I 


d1A1  +  4dm  X  m).    (27) 


FIELD-MAXUAL    FOK    ENGINEERS. 
OVERHATTL. 

No  allowance  is  made  for  moving  excavated  material  when  the 
haul  does  not  exceed  a  certain  specified  distance  culled  the 
"  limit  of  free  haul."  But  when  the  material  is  carried  beyond 
this  limit  the  extra  labor  involved  is  paid  for  at  a  stipulated 
price  per  cubic  yard,  per  each  100  feet  in  excess  of  the  tree-haul 
limit. 


0  b         f/ 


FIG.  144. 

Let  us  suppose  the  material  in  the  cut  cO,  Fig.  144,  just 
sufficient  to  make  the  fill  Od. 

First  find  on  the  profile  two  points  a  and  b  such  that  the  cut  aO 
will  just  make  the  fill  Ob,  and  that  the  distance  ab  is  equal  to  the 
limit  of  free  haul. 

These  points  a  and  b  are  found  by  means  of  the  cross-sections 
and  calculated  quantities  ;  though  since  a,  as  well  as  b,  usually 
falls  between  regular  stations,  it  is  generally  necessary  to  find  the 
point  a  by  one  or  more  trials,  so  that  ab  is  equal  to  the  required 
limit. 

Having  found  ab,  it  is  to  be  remembered  that  the  contractor  is 
entitled  to  pay  for  moving  every  cubic  yard  of  material  from  the 
cut  ca  to  the  fill  bd'  for  the  whole  distance  it  is  moved,  less  the 
distance  ab. 

This  is  equivalent  to  moving  the  whole  cut  a  distance  ggr  —  ab 
=  ga  -f  bg' ;  g  and  g'  being  respectively  the  centers  of  gravity  of 
the  cut  ca  and  of  the  fill  bd  made  from  it. 

But  the  volume  of  ac  multiplied  by  the  distance  ag  is  equal  to 
the  sum  of  the  products  obtained  by  multiplying  the  volume  of 
each  prismoid  in  ac  by  the  distance  of  its  own  center  of  gravity 
from  «. 

It  is  usually  sufficiently  accurate  to  consider  the  center  of  grav- 
ity of  a  prismoid  as  being  at  its  mid -section;  but  if  greater  accu- 
racy is  required,  we  have  for  the  distance  from  the  mid-section 

x  —  —  .  —     — -,  very  nearly.      ,     .     .     o     (28) 
6      A  -\-  A. 


CALCULATION   OF   EARTHWORK.  2G9 

This  is,  of  course,  to  he  added  to  or  subtracted  from  the  dis- 
tance of  the  mid-section  from  the  point  a  accordingly  as  the 
larger  end  area  A  is  the  farthest  from  or  nearest  to  the  point  a. 

In  the  same  manner  we  find  the  sum  of  the  products  obtained 
by  multiplying  the  volume  of  each  prisinoid  in  bd  by  the  distance 
of  its  center  of  gravity  from  b. 

Summing  the  products  for  both  cut  and  fill,  the  distances  be. 
ing  expressed  in  chains  of  100  feet,  and  multiplying  the  result  by 
the  stipulated  allowance,  we  have  the  amount  to  be  paid  for  haul. 

Parts  of  a  cut  may  be  carried  in  opposite  directions,  in  which 
case  each  part  must  be  figured  separately. 

It  is  evident  that  no  allowance  is  made  on  material  wasted. 

MONTHLY  ESTIMATES. 

Monthly  estimates  are  usually  made  by  the  resident  engineer 
near  the  end  of  each  month,  though  it  is  sometimes  necessary  to 
make  them  at  other  times.  For  this  purpose  calculated  quanti- 
ties in  the  field-books  are  used,  supplemented  by  such  measure- 
ments as  may  be  necessary. 

The  monthly  estimate  is  only  approximate,  and  should  not  be 
above  the  real  amount  of  work  done;  but  it  should  be  as  definite 
and  complete  as  the  nature  of  the  case  will  permit,  and  should 
include  a  detailed  statement  of  all  the  work  done  and  material 
delivered. 

A  special  field-book  is  devoted  to  monthly  estimates.  Cross- 
sections  must  be  taken  for  the  purpose  of  computing  the  amount 
of  excavation  completed  ;  and  notes  of  everything  done  must  be 
made. 

Where  work  is  completed,  the  corresponding  quantities  may  be 
taken  from  the  field-books  or  other  books  containing  them. 

An  allowance  somewhat  below  the  actual  value  should  be  made 
for  all  material  delivered  but  not  yet  used,  as  well  as  for  all  labor 
performed  and  expenditures  made  in  properly  forwarding  the 
work. 

Estimate  sheets  should  be  used,  the  sheet  for  each  month  show- 
ing, for  different  parts  of  the  work  and  for  each  kind  of  material, 
the  total  of  the  previous  estimates,  the  present  estimate,  and  the 
total  including  the  present  estimate. 

The  total  estimate  for  any  month  becomes  the  total  "previous 
estimate"  for  the  succeeding  month. 


270 


FIELD-MANUAL   FOB    ENGINEERS. 


The  division  engineer,  or  the  chief  engineer  in  case  there  is 
no  division  engineer,  reviews  these  reports,  copies  them  on  other 
sheets;  attaches  the  prices  to  the  items  ;  computes  and  sums  up 
the  amounts. 

The  railway  company  pays  the  contractor,  each  month,  about 
85$  of  the  estimate  for  that  month,  and  retains  the  rest  till  the 
completion  of  the  contract. 

FINAL  ESTIMATE. 

The  final  estimate  is  a  complete  statement  in  detail  of  all  the 
work  done  and  of  all  the  material  furnished  by  the  contractor, 
and  furnishes  the  basis  of  final  settlement  between  the  company 
and  the  contractor. 

This  statement  is  completed,  in  detail,  as  the  work  progresses. 
As  soon  as  the  data  of  any  part  or  subdivision  of  the  work,  or  of 
any  structure,  are  supplied,  a  complete  statement  in  regard  to  it 
should  be  written  out  in  detail  in  a  book  for  that  purpose.  The 
number  of  cubic  yards  in  each  prismoid,  extra  for  cutting  sur- 
face ditches,  overhaul,  etc.,  should  be  given.  A  complete  state- 
ment in  regard  to  each  bridge  or  other  structure  should  be  cure- 
fully  written  out.  The  notes  should  be  made  especially  full, 
while  the  work  is  in  progress,  in  regard  to  all  parts  of  it  which 
are  inaccessible  after  completion,  such  as  foundation-pits  and 
foundations  of  all  kinds,  and  all  works  under  water. 

COMPUTATION  OF  PRISMOIDS — LEVEL  LATERALLY. 

Let  abmn  (Fig.  145)  represent  one  end  of  an  embankment,  100 
feet  in  length,  level  laterally. 


FIG.  115. 

Let  cd,  ef,  etc.,  represent  horizontal  planes  which  divide  the 
embankment  into  layers  0.1  of  a  foot  in  height. 


CALCULATION    OF    EARTHWORK. 


271 


Let  the  base  ab  =  b,  and  tlie  side  slope  ——==».      Then 

a  I) 

ab  =  b; 

cd  —  b  +  0.2*; 

«/=  &  +  0.4s,  etc. 

** 


Also, 
Hence 


+  ** 


=  5  + 


=  &  +  0.3*.  etc. 


the  vol.  in  yards,  of 


=       -  +  0.1.)  = 


27 


and 


vol  of  cdef  =  ^(b  +  0.3*)  = 


etc. 


It  is  plain  tliat  the  volumes  of  tlie  successive  layers  are  in  aritli- 

2.9 

metical  progression,  Laving  a  common  difference  of  —  .     Hence  to 

find  the  total  volumes  corresponding  to  different  heights  of  the 
embankment,  differing  by  0.1,  it  is  only  necessary  to  write  down 

for  the  volume  of  abed:  add  -  —  -  —  to  it  for  the  volume 


of  abef,  then 


to  the  last  result  for  the  volume  of  abgh, 


etc.,  omitting,  for  convenience,  the  denominator  27. 
Example  1.—  Let  b  =  18,  and  8=1.     Then 


~-  =  6  +  if 
Hence  we  have  as  follows  : 


27 


etc. 


Heights. 

Volumes. 

Heights. 

Volumes. 

0.1 

6          19 

6        21 

0.4 

27.          7 
7          0   • 

0.2 

13        13 
6        23 

0.5 

34          7 

7          2 

0.8 

20          9 
6        25 

0.6 

41          9 
etc.,  etc. 

• 

ElELD-MANUAL    FOR    EXGIXEE!^. 


This  may  easily  be  carried  20  or  30  feet  in  an  hour,  or  less, 
Example  2. — Let  b  =  14,  and  s  =  f .     Then 


27 
Then  as  follows  : 


Heights. 

Volumes. 

Heights. 

Volumes. 

0.1 

5        13 
5         19 

0.5 

27         17 
5        43 

0.2 

10        32 
5        25 

0.6 

33          6 
5        49 

0.3 

16          3 
5        31 

0.7 

39          1 
6          1 

0.4 

21        34 
5        37 

0.8 

45          2 

etc.,  etc. 

These  results  can  be  copied  to  the  nearest  yard  very  quickly. 

The  author  hopes  to  show  the  best  processes  of  computing-  all 
earthwork  and  other  tables,  and  of  making  all  numerical  com- 
putations,  in  the  near  future. 


CHAPTER   XI. 
APPROXIMATE  AND  ABRIDGED   COMPUTATIONS. 

THIS  chapter  will  treat  of  the  subject  of  approximate  -rind 
abridged  computations  to  a  very  limited  extent  only,  dealing 
mainly,  too,  with  the  practical  side  of  the  question. 

Approximate  computations  are  those  that  lead  to  results  not 
strictly  accurate. 

Abridged  computations  are  those  that  reach  certain  resuliswith 
lef--s  labor  than  the  ordinary  computations  involve.  Abridged 
computations  may  be  either  accurate  or  approximate,  but  are 
usually  the  latter.  The  absolute  error  of  a  result  is  the  difference 
between  that  result  and  the  true  result  for  which  it  stands. 

The  relative  error  of  a  result  is  the  absolute  error  dividf  d  by 
the  true  result,  and  it  is  therefore  equal  to  the  absolute  error  of 
each  unit  of  the  result. 

Thus  if  a  true  result  is  84  and  the  approximate  value  77  is 
taken  for  it,  the  absolute  error  is  84  —  77  =  7,  and  the  absolute 

7  1 

error  of  each  unit  of  the  result,  or  the  relative  error,  is-^-  =  — — . 

o4  1/i 

It  is  important  to  notice  that  the  relative  error  of  a  result  is 
not  changed  by  multiplying  that  result  by  any  number;  since 
ths  error  in  the  product  or  quotient  will  be  increased  or  decreased 
in  the  same  ratio  as  the  result  is  increased  or  decreased. 

Thus  if  n  be  the  true  and  n  -f-  e  the  approximate  result,  then 

r  —  -  is   the   relative   error.     Multiplying   by  m,  we   have   the 

Ti 

approximate  product  m(n  -j-  e}  —  mn  -f  me,   whereas    the   true 
product  is  mn.     Hence  the   absolute  error  of  the  product  is  me, 

and  the  relative  error  is  =  —  as  before. 

mn       n 

If  an  approximate  quantity  is  given,  the  absolute  error  of 
which  is  a  given  number  of  units  of  a  given  order  counting  from 

373 


274  FIELD-MANUAL   FOR   ENGINEERS. 

the  highest,  it  follows  from  tlie  preceding  that  the  relative  error 
is  independent  of  the  decimal  point.  Thus  suppose  the  absolute 
error  of  each  of  the  numbers  7.073,  .07073,  and  707.3  to  be  two  units 
of  the  fourth  order.  The  left-hand  significant  figure  expresses 
units  of  the  first  order,  and  therefore  the  fourth  order  is  express  .-d 
in  the  above  by  the  figure  3.  The  relative  errors  of  the  numbers 

.002      .00002  .2  2 

are  T073~'  :07073'  an<L  -^P  eaCU  °f  wlilcL  1S  equal  toTo73' 
Hence  if  a  number  is  given  whose  absolute  error  does  not 
exceed  a  given  number  of  units  of  any  order  (the  »th,  say),  an 
approximate  and  convenient  limit  of  the  relative  error  is  readily 
found  by  dropping  all  tLe  figures  at  the  right  of  the  ?ith  figure, 
replacing  with  zeros  all  the  others,  except  one  or  two  at  the  left, 
and  dividing  the  given  number  of  units  by  the  result,  regarded  as 
a  whole  number.  Thus  we  know  that  it  —  3.1415  to  within  less 
than  a  unit  of  the  fifth  order  or  fourth  decimal  place,  and  hence  the 

,     .0001  1  1  1 

relative  error  cannot  exceed  — —  — -  =  — — :,  or  •   ,  ^..  or 

3.1415        3141o        30000         31000 

etc. 

The  first  n  highest  figures  of  an  approximate  number  express 
the  value  of  that  number  to  n  places.  The  same  figures,  in- 
creasing the  last  by  unity,  when  the  following  figure  is  5  or  more, 
express  the  exact  value  to  n  places  of  the  same  number.  Thus 
3.141592  being  an  approximate  value  of  n,  3.1415  is  called  the 
value  of  it  to  five  places,  and  3.1416  is  called  the  exact  value  of  it 
to  five  places. 

ADDITION. 

To  find  the  relative  error  of  the  sum  of  any  number  of  approx- 
imate quantities. 

Let  HI  ,  7i2  •  •  •  nm  represent  the  true  values  of  the  quantities, 
and  ei ,  e-i  .  .  .  em  represent  their  absolute  errors.  Let  8  represent 
the  sum  of  the  quantities,  and  EJt\ie  sum  of  the  absolute  errors. 

Then 

E  —  d  -f  62  -f  .  .  .  em,     S  —  ni  -f  n^  -f  .  .  .  nm, 
and  the  relative  error  is 


APPKOXjUMATK    A-N  V 


CO31FU  J  ATiOJNS.     %  <  O 


Supposing  the  errors  to  be  on  the  same  same  side,  then,  in  case 
they  are  equal,  the  relative  error  of  the  sum  of  the  quantities  is 
equal  to  the  relative  error  of  each  of  the  quantities  ;  but  when 
they  are  unequal,  th's  error  of  the  sum  is  greater  than  the 
smallest  relative  error  of  the  quantities,  and  smaller  than  the 
greatest  of  such  errors.  If  the  errors  are  on  opposite  sides  of  the 
•..Tilth,  the  relative  error  of  the  sum  must  be  less  than  the  greatest 
relative  error,  and  may  be  equal  to  zero. 

It  is  sufficient,  therefore,  in  practice  to  take  each  number  to  be 
added  as  accurate  as  it  is  desired  that  the  result  shall  be. 

Example. — Find  the  sum  of  the  square  roots  of  the  numbers 
2,  3,  5,  6,  7,  8,  10,  11,  12,  u.ul  13  true  to  about  the  fifth  place. 
Taking  the  square  roots  exact  to  five  places,  we  have  the  follow- 
ing result  : 

1.4142  + 
1.7321  - 
2.2361  - 
2.4495  _ 

2.6458  - 
2.8284  + 
3.1623  - 
3.3166  + 
3.4641  + 
3.6056  - 


26.8547 


Without  examination  we  know  from  the    preceding  that  the 

relative  error  of  this  result  must  be  less  than  - 

14143       28284 

which  is  less  than  .     Hence  the  absolute  error  of  the  result 


must  be  less  than  a  unit  of  the  fifth  place,  or  less  than  .001.  Since, 
however,  there  are  only  ten  numbers  added,  the  error  in  the 
right-hand  or  sixth,  figure  of  the  result  cannot  exceed  ^  X  10  =  5, 
and  the  error  in  the  fifth  figure  cannot  exceed  5  -j-  10  =•  \. 
Furthermore,  there  are  only  six  errors  in  the  same  direction,  and 
hence  the  error  in  the  fifth  place  cannot  exceed  i  X  6  -?-  10  =  -.,*,- 
of  a  unit. 


UNIVERSITY 

OF 


276  FIELD-MANUAL   FOR   ENGINEERS. 


SUBTRACTION. 

Let  HI  and  7i2  represent  two  quantities,  nl  being  greater  than 
«.2  ;  and  let  n^  -\-  e\  and  «a  -\-  e2  represent  approximate  values 
of  the  same.  The  true  difference  is  n\  —  7i8;  the  difference  be- 
tween the  approximate  values  is 

fti  +  «i  —  (nt  ±  0a)  =  (w,  —  na)  -f  ti  T  «a. 

Hence  0,   ±  e9  is  the  absolute  error  of  the  difference  due  to  errors 
in  the  numbers. 

The  relative  error  of  the  difference  is 


Til   — 


The  minuend  and  subtrahend  are  usually  taken  true  to  ^  of  a 
unit  of  the  last  place,  in  which  case  the  absolute  error  of  the  dif- 
ference cannot  exceed  a  unit  of  the  same  place  or  order  of  units. 

The  relative  error  of  two  numbers,  nearly  equal,  may  be  quite 
large,  as  shown  in  the  following  example; 

Compute  |/83  —  |/18  correct  to  five  places  and  find  the  relative 
error  of  the  result. 

|/83  =  4.3621  to  within  ^  a  unit  of  the  fifth  place. 

1/18  —  4.2426  to  within  i-a  unit  of  the  fifth  place. 
Difference  =  0.1195  to  within  1  unit  of  the  fourth  place. 

The   relative   error   of   either   of   the   numbers    cannot   exceed 
84ft  59'  w^^e  ^ue  relative  error  of  the  difference  may  be 


nearly  -J-T^:,  which  is  71  times  as  large  as  the  former. 


Any  degree  of  accuracy  may  be  secured  in  the  result  or  differ- 
ence, however,  by  taking  the  numbers  sufficiently  accurate. 


APPROXIMATE   AND   ABRIDGED   COMPUTATIONS.    277 


MULTIPLICATION  AND  DIVISION. 

To  find  the  error  in  a  product  due  to  errors  in  one  or  both 
factors. 

Let  nl  and  ;?2  represent  any  numbers,,  HI  -f-  f-\  and  n-^-\-f-t  ap- 
proximate values  of  the  same  as  above. 

Let  N  =  WiWa,  -ZVi  =  (H-I  +  «i)(wa  H-  £3),  and  let  E  =  Nt  —  N  — 
the  absolute  error  of  the  product. 

Now  JVj  —  JV=  7J.)da  +  #2*1  -f-  tfi^a  =  n\e*  -f-  n2e1,  very  nearly, 
and  therefore 


E 


or,  generally,  -—  =  —  ±  — (4) 

.ZV  yij  72/2 


The  two  results  given  by  eq.  (4)  occur  equally  often.  Hence 
the  relative  error  of  the  product  of  two  numbers  can  never  exceed 
the  sum  of  the  relative  errors  of  the  numbers,  and  it  is  as  often 
equal  to  the  difference  of  such  errors  as  to  their  sum. 

If  ei  =  0, 


Hence  if  only  one  factor  is  in  error,  the  relative  error  of  the 
product  is  equal  to  the  relative  error  of  the  factor  in  error. 

If  e\  and  0a  are  in  opposite  directions,  and  —  —  —  ,  then 


(6) 


Hence  two  numbers  may  be  changed  in  opposite  directions  and 
in  proportion  to  their  magnitudes  without  sensibly  affecting  their 
product.  Thus  41  X  82  =  3362,  and  40  X.84  =  3360. 


278  FIELD-MANUAL    FOR   ENGINEERS. 

Comparing  multiplication  with  division,  we  know  that  the 
product  in  the  former  corresponds  to  the  dividend  in  the  latter, 
and  the  factors  in  the  former  to  the  divisor  and  the  quotient  in 

ff 
the  latter.       Hence  —   may  represent   the   relative   error   of  the 

dividend,  and  —  and  —  such  errors  in  the  divisor  and  the  quo- 
tient. Transposing  eq.  (4),  therefore,  and  we  have 

^  -  -  T  ±  (7) 

m  ~  N       n,' 

This  shows  that  the  relative  error  of  the  quotient  (or  divisor)  is 
equal  to  the  difference,  or  to  the  sum,  of  the  relative  errors  of  the 
dividend  and  divisor  (or  quotient). 

It  follows  that  when  the  divisor  is  correct  the  relative  error  of 
the  quotient  is  equal  to  that  of  the  dividend,  and  when  the  divi- 
dend is  correct  the  relative  error  of  the  quotient  is  equal  to  that 
of  the  divisor,  but  with  a  contrary  sign. 

Furthermore,  the  divisor  and  the  dividend  may  be  both  in- 
creased or  both  decreased,  in  proportion  to  their  magnitudes, 
without  affecting  the  quotient. 

If  ni,  n9  .  .  ,  nm  represent  factors,  and  TV  their  product,  etc. ,  then 


(8) 


e\,  e$,  etc.,  may  be  either  positive  or  negative,  and  hence  the  rela- 
tive error  of  the  product  is  equal  to  the  algebraic  sum  of  the  rela- 
tive errors  of  the  factors. 

Without  here  discussing  the  average  value  of  the  relative  error 
of  the  product,  it  is  sufficient  to  state  that  it  rarely  much  exceeds 
the  greatest  relative  error  in  the  factors  ;  and  hence  the  factors 
only  need  be  taken  to  about  that  degree  of  accuracy  desired  in 
the  product.  Similarly,  the  divisor  and  the  dividend  should  be 
taken  to  about  that  degree  of  accuracy  desired  in  the  quotient. 

Example.—  Find  the  product  89.0245  X  .0194525  true  to  about 
four  places. 

89.02  X  .01945  =  1.731439,  true  to  less  than  \  a  unit  of  the 
fourth  place, 


Suppose  we  have,  for  example,  .ZV  =abc~}-  def -{-  gh,  and  sup- 
pose the  relative  errors  in  the  factors  are  each  equal  to  e. 

Then  it  may  be  shown  that  the  relative  error  of  the  product 
cannot  exceed  the  relative  error  of  that  term,  which  contains  as 
many  factors  as  any.  In  the  expression  above,  the  relative  error 
of  N  could  not  exceed  the  relative  error  of  the  term  abc  or  def,  or 
-three  times  the  common  relative  error  of  the  factors,  or  3e. 

Suppose  Q  =  — —  — .     Then  the  relative  error  of  Q  canr.ot 

fClr    "~p"     77i 

exceed  the  greatest  relative  error  of  any  term  in  the  numerator, 
plus  the  greatest  relative  error  of  any  term  in  the  denominator  ; 
or  the  relative  error  of  a  term  equal  to  the  product  of  such  terms. 
These  terms  in  the  above  are  abc  and  kl,  and  hence  the  relative 
error  in  Q  cannot  exceed  that  of  abckl,  or  5e. 

Since  the  errors  of  the  factors  are  usually  quite  unequal  and  on 
opposite  sides  of  the  truth,  the  relative  error  of  the  product  or  of 
the  quotient  rarely  much  exceeds  the  greatest  relative  error  o!  the 
factors;  and  hence  in  all  cases  it  is  practically  sufficient  to  take 
the  factors  to  about  that  degree  of  accuracy  desired  in  the  result. 


ABRIDGED  MULTIPLICATION. 

To  find  the,  product  of  two  factors  true  to  a  unit  of  the  nth 
order. 

Rule. — Write  the  multiplier  in  the  inverse  order,  placing  the 
highest  order  under  the  nth  order  of  the  multiplicand.  Multiply 
by  each  figure  of  the  multiplier,  beginning  with  the  figure  in  the 
multiplicand  above  it,  rejecting  the  part  of  the  multiplicand  to 
the  right  of  it,  except  to  carry  from,  and  place  the  right-hand 
figures  of  the  partial  products  under  each  other.  Annex  at  the 
right  of  the  partial  product  a  number  of  zeros,  equal  to  the  num- 
ber of  figures  in  the  multiplier  and  multiplicand  less  n  -|  •  1,  and 
point  off  from  the  right  for  decimal  factors  as  usual. 

With  respect  to  products  containing  decimals,  the  decimal  point 
is  easily  determined  by  inspection,  and  in  this  case  the  zeros  need 
not  be  added. 

Examples.—  Find  615694  X  59019,  367847  X  278437,  and 
3.141592  x  1.4142,  probably  true  to  a  unit 
of  the  nth  or  fourth  order, 


280  FIELD-MANUAL   FOB   ENGINEERS. 

Arranging  the  factors  and  multiplying  as  directed,  we  find 

615694  367847  3.141592 

91095  734872  24  141 


30785  7357  3  142 

5541  2575  1  257 

6  294  31 

6  15  12 


36338000000  102420000000  4.443 

With  respect  to  the  above,  we  observe  that,  usually,  as  in  the 
first  and  second  examples,  the  right-hand  figures  of  the  partial 
product  are  of  the  n  -f-  1  (fifth  in  this  case)  order,  and  that  the 
error  in  each  of  these  figures,  if  the  carrying  has  been  attended  to, 
cannot  exceed  ^  of  a  unit,  or  2£  units  in  all,  or  2£  -5-  10  =  £  of  a 
unit  of  the  fourth  order.  For  reasons  already  given,  the  error 
will  rarely  exceed  1  unit  of  the  n  -f-  1  order. 

When  the  highest  figures  of  the  .factors  are  sufficiently  small, 
as  in  the  third  example,  so  that  there  are  only  n  figures  iu  any  of 
the  partial  products,  and  also  in  their  sum,  the  right-hand  figures 
of  the  partial  products  will  be  of  the  nth  order,  and  it  is  possible 
for  the  result  to  be  in  error  more  than  a  unit  of  the  nih  order, 
though  such  an  error  is  quite  improbable. 

In  this  case  we  observe,  also,  that  the  left-hand  figures  of  the 
products  are  always  larger  than  those  of  either  factor,  so  that  a 
possible  error  of  over  a  unit  in  the  nth  place  of  the  result  is  not 
of  much  practical  consequence.  To  make  sure  of  avoiding  this 
possible  error,  the  right-hand  figure  of  the  inverted  multiplier 
must,  in  this  case,  be  placed  under  the  n  -f-  1  figure  of  the  multi- 
plicand. 

To  find  the  product  of  any  number  of  factors  true  to  about  a 
unit  of  the  nth  place,  proceed  with  each  product  by  the  above 
rule. 

Example. — Find  the  product,  true  to  about  a  unit  of  the  fourth 
place,  of  762.8314  X  6.821426  X  4827.31  X  .027265. 


APPROXIMATE    AND    A  15  RI  DUE  I)    COM  I'l'TATIONS. 

762.88       5203.6       25119000 
4128  6       87284        56272 


4577  0  20814  50238 

610  2  4163  17583 

15  3  104  502 

8  36  151 

3  2  13 


5203.6       251.19000       684870 

The  right-hand  figures  of  the  multipliers  are  placed  under  the 
;*th  (fourth)  figure  of  the  multiplicand,  except  in  finding  the  last 
product,  where  it  is  placed  under  the  n  -}-  1  (fifth),  so  that  there 
will  be  ;/  -)-  1  figures  in  each  product.  The  number  of  integral 
figures  in  each  product  is  easily  found  by  inspection. 


DIVISION. 

A  divisor  and  a  dividend  being  given  to  any  required  degree  of 
exactness,  to  find  the  quotient  true  to  about  a  unit  of  the  /ith 
order. 

Mule. — Take  the  divisor  to  n  significant  figures  and  enough  exact 
figures  in  the  dividend  (n  or  n  -f-  1)  to  contain  this  divisor.  Mul- 
tiply the  divisor  by  the  first  quotient  figure,  and  subtract  the 
product  from  the  assumed  dividend;  and  instead  of  annexing  any 
figure  to  the  remainder,  reject  the  right-hand  figure  of  the 
divisor  to  determine  the  second  quotient  figure.  Continue  the 
process  of  division  by  rejecting  successively  the  right-hand  figures 
of  the  divisor  until  the  quotient  contains  n  figures. 

In  multiplying  by  the  first  quotient  figure  carry  from  the  part 
of  the  divisor  at  the  right  of  the  nili  figure,  and  in  each  multipli- 
cation carry  from  the  part  of  the  divisor  at  the  right  of  the  part 
multiplied,  so  as  to  make  each  partial  product  true,  if  possible,  to 
within  ^  a  unit  of  the  lowest  order. 

Example.—  Find  the  quotient  of  1497.82746  divided  by  16.72374 
to  within  about  a  unit  of  the  fifth  order.  Place  the  divisor  on  the 
riglit  of  the  dividend,  and  the  quotient  under  the  divisor,  and 
cross  out,  when  used,  the  successive  right-hand  figures  of  the 
divisor.  '1  hus  we  have 


FIELD-MANUAL    FOR    ENGtKEEHS. 

149783)1«?&874 
1 33790  89563 


1st  remainder     15993 
15051 

3d          "  942 

836 


3d  106 

100 


(n  —  l)tli       "  6 

5 


nth 


The  last  partial  product  (5  in  the  example)  and  the  last  re- 
mainder  (1  in  the  example)  need  not  in  practice  be  written. 

To  determine  the  possible  error  in  tbe  quotient,  Ave  will  find  the 
possible  error  in  the  last  remainder  used  in  finding  the  quotient. 

Now  the  error  in  the  dividend  cannot  exceed  I  a  unit  of  tbe 
lowest  order,  and  the  error  in  each  partial  product  cannot  exceed 
-£  a  unit  of  its  lowest  order,  which  is  the  same  as  that  of  the 
dividend.  The  error  in  the  (n  —  l)th  remainder  (6  in  the  ex- 
ample) cannot  exceed  the  sum  of  these  errors,  or 


+  i  X  (n  -  1)  -      . 


The  nth  or  last  quotient  figure  (3  in  the  example)  is  found 
by  dividing  the  (n  —  l}th  or  next  to  the  last  remainder  (6  in 
the  example)  by  the  first  figure  in  the  divisor  (having  regard 
to  currying  from  the  other  figures),  and  cannot  exceed  that 
remainder.  But  there  is  no  error  in  the  divisor,  and  hence  the 
relative  error  of  the  last  quotient  figure  (and  hence  of  the  quotient) 
cam.  ot  exceed  the  relative  error  of  next  to  the  last  remainder;  and 
since  the  last  quotient  figure  cannot  exceed  the  next  to  the  lowest 
remainder,  the  absolute  error  of  the  quotient  cannot  exceed  the 

absolute  error  of  that  remainder,  or  -  . 

2 

It  is  plain  that  the  error  in  question  is  usually  less  than  unity. 


APPROXIMATE    AND   ABRIDGED   COMPUTATIONS.     283 

Example  1. — Find  the  quotient  — with  an  absolute  error 

763054 

in  the  quotient  less  than  a  unit  of  the  sixth  place. 

1941690)763054 
1526108  254463 

Tl^582 
381527 


34055 
30522 

3533 
3052 

481 
458 


Example  £.—  Compute  the  expression  —  =  with  a  relative  error 

of  less  than  .0001. 

We  easily  see  that  the  quotient  will  be  greater  than  2,  and  hence 
we  may  make  an  absolute  error  of  .0002.  It  will  be  more  than 
sufficient,  then,  to  find  the  quotient  true  to  a  unit  of  the  fourth 
decimal  place. 

Thus  we  find 

3.  1415)1.  4J42 
282842.2214 


3131 


303 
283 

~20 
14 


CHAPTER  XII. 

CONSTRUCTION. 

CLEARING  AND  GRUBBING. 

THE  first  operation  in  constructing  a  railway  is  to  clear  off  all 
timber  upon  the  right  of  way.  The  engineer  should  provide  in 
his  specifications  for  the  cutting  into  proper  lengths,  and  storing, 
all  valuable  timber  near  the  borders  of  the  right  of  way.  The 
refuse  should  be  burned. 

Where  a  deep  cut  is  to  be  made,  the  trees  are  felled  and  the 
stumps  removed  as  the  earth  is  excavated,  and  the  stumps  are 
covered  up  in  deep  fills.  In  very  shallow  cuts  and  fills  it  is  best 
to  pull  over,  and  out,  the  trees  by  their  roots,  rather  than  to  fell 
them  and  afterwards  to  grub  out  the  stumps,  though  this  is  often 
done. 

Since  in  fills  the  tops  of  stumps  should  be  at  least  1|  feet  below 
grade,  it  is  easy  to  tell  the  trees  that  can  best  be  cut  down,  and 
those  that  must  be  grubbed  out. 

The  amount  of  grubbing  required  being  usually  difficult  to 
estimate  accurately,  it  is  decidedly  the  best  way  to  make  no  sep- 
arate account  of  it,  but  to  include  it  in  the  earthwork. 

GRADE  LINE. 

Having  a  profile  of  the  line,  a  grade  line  is  established  thereon, 
first  tentatively  by  the  eye,  and  then  definitely  determined  by 
computation,  so  as  to  make  the  cost  of  the  earthwork  a  minimum, 
due  regard  being  had  to  total  cost  and  controlling  conditions,  such 
as  the  height  of  bridges,  road-crossings,  etc.,  and  especially  drain- 
age. 

The  natural  waterways  must  be  crossed  high  enough  to  afford 
sufficient  openings  for  the  flow  of  the  water. 

284 


CONSTRUCTION".  28fi 

In  order  to  form  a  correct  idea  as  to  tlie  size  of  openings  and 
culverts  needed,  the  engineer  must  observe  the  water-marks  on 
trees,  etc.,  note  the  width  and  slope  of  valleys,  and.  if  need  be, 
measure,  approximately,  the  area  of  watersheds  adjacent  to  the 
line.  It  is  perhaps  needless  to  note  what  every  engineer  of  ex- 
perience knows,  that  much,  property,  and  some  lives,  are  often 
lost  by  neglecting  these  precautions.  It  is  well  to  observe,  too, 
that  the  concentration  of  water  in  drainage  ditches  very  much  in- 
creases the  flow;  since  when  it  is  widely  spread  out  over  marshes, 
meadows,  etc.,  much  of  it,  or  all  of  it,  may  soak  into  the  soil  or 
evaporate. 

The  concentration  of  small  streams  by  means  of  side  ditches, 
thus  avoiding  unnecessary  culverts,  is  often  economical. 

Ample  side  ditches  should  be  constructed  to  keep  the  water  off 
the  roadway. 

Surface  ditches  should  also  be  made  along  the  line,  a  few  feet 
from  the  upper  edge  of  cuts,  to  prevent  the  surface-wa1er  from 
pouring  into  these  after  rains.  Neglect  of  this  causes  much  un- 
necessary labor,  and  sometimes  leads  to  accidents. 

It  is  very  desirable  to  substitute  light  fills,  in  place  of  cuts,  so 
far  as  practicable,  since  in  cuts  it  is  difficult  to  maintain  a  good 
track. 

Having  established  the  grade  line,  the  difference  between  its 
height  and  that  of  the  surface  of  the  ground  at  different  points 
gives  the  cuts  or  fills  at  these  points,  along  the  center  line.  These 
should  be  plainly  marked  on  the  center  stakes  in  feet  and  tenths. 
Moreover,  the  heights  of  the  ground  at  or  near  all  points  between 
stations,  where  it  changes  much,  should  be  found,  and  the  corre- 
sponding cut  or  fill  marked  on  stakes  set  for  the  purpose.  It  is 
well  to  observe  that  cross-sections  may  be  chosen  between  such 
points,  thus  diminishing  their  number,  but  giving  and  taking  so 
as  to  determine  the  true  volume  as  accurately  as  though  all  the 
points  were  used. 

A  cross-section  should  be  taken  not  only  where  the  irregulari- 
ties of  the  center  line  require  it,  but  also  where  the  irregularities 
on  either  side  may  require  it,  the  object  being  to  delineate  the 
outline  of  the  earthwork,  and  also  to  furnish  the  data  for  an  accu- 
rate computation  of  the  same.  A  "  berm  "  5  or  6  feet  wide  should 
be  left  between  the  edges  of  all  side  ditches  and  the  foot  of  the 
adjacent  fill. 


286 


FIELD-MANUAL   FOR    ENGINEERS. 


STAKING  OUT  EARTHWORK. 

A.  When  the  ground  is  level  laterally,  as  shown   in   Fig.  146, 
the  operation  is  very  simple. 


FIG.  146. 

Let  2&  =  the  width  of  the  roadbed  ab,  and  c  —  CCi,  the  center 
height.  The  side  heights  P'm'  and  H'n'  are  also  equal  to  c.  Let 
8  —  the  slope  of  the  side  P'a  or  H'b.  Then 

bn'  =  H'n'  X  8  =  c.8,     and     CH'  =  d  =  b  -f  r& 

This  shows  the  distance  out  from  C  to  7/',  where  the  excavation 
must  begin,  in  order  that  the  foot  of  the  slope  at  b  may  be  at  the 
proper  distance  db  =  b  from  the  center  C\. 

If  in  cross-sectioning,  whether  level  laterally  or  not,  the  heights 
of  additional  points  between  stations  are  required,  they  must  be 
found. 

It  is  also  necessary  to  find  the  point  where  the  grade  line 
intersects  the  surface  of  the  ground  ;  since  at  that  point  there  is 
neither  cut  nor  fill. 

A  stake  is  set  there  and  marked  0.0. ;  and  the  corresponding 
side  stakes  are  set  and  marked.  Also  the  points  where  the  edges 
of  the  proposed  roadbed  intersect  the  surface  of  the  ground  are 
found,  and  0.0.  is  marked  on  the  stakes  set  at  those  points. 
Cross-sections  are  taken  at  these  points  also. 

Let  Fig.  147  represent  the  case  in  which  the  ground  is  not  level 
laterally.  Instead  of  the  cross-section  abP'H' ',  as  in  the  former 
case,  we  now  have  the  cross-section  abPGA,  PGA  being  the  sur- 
face of  the  ground. 

We  will  first  show  how  to  find  the  position  of  the  side  stakes 
by  computation,  where  the  slope  of  the  ground  is  uniform  from 


CONSTRUCTION. 


28? 


the  center  stakes  to  the  side  stakes  ;   not   for  the  purpose  of  act- 
ually locating  the  stakes  in  this  way,  for  the  ground  rarely  slopes 


P'diL 


m'    in 


FIG.  147. 


uniformly  in  any  direction,  (and  therefore  this  could  not  be  done 
with  much  accuracy). 

It  will,  however,  aid  in  showing  the  effect  of  the  slope  of  the 
ground  in  cross-sectioning,  as  we  shall  see. 

Let  CH'  =  d,  H'K  =  d',  CK=d  +  d'=  D;  8  =  the  side  slope, 
and  8'  -  the  slope  of  the  ground.  Also,  LP'  =  dt,  GL  =  E. 


Then 

and 

Dividing  gives 

from  whicli 


8  = 


o/  


df 
AK' 

d  +  d' 
AK  ' 

d'  +  d 
d'     ' 

dS 


8'  -  8  ~ 
Adding  CH'  =  d  to  the  preceding  gives 

CK=  D  =  y  __ti  =      £,  _  ^ 


(1) 


(2) 


288  FIELD-MANUAL    FOR    ENGINEERS. 

Dividing  (1)  by  CH'  =  d  gives 

d  _  11'  K  _        8 

d  ~  CU'   ~  8'  -  S  .....     •     •     (3) 

Dividing  (1)  by  (2)  gives 

&      IVK        St 


_ 

CK   ~~~  8' 


These  equations  enable  one  to  judge  quite  accurately  the  addi- 
tional distance  beyond  H'  necessary  to  go  out  on  the  upper  side 
of  a  cut  in  consequence  of  the  slope  of  the  ground. 

Thus  if         8=  ^,     d'  =  J;     if     8  =  ?-',     d'  =  -,     etc. 
4  o  54 


Eq.  (3)  shows  that  the  additional  distance  UK  is  to  the  distance 
CH',  for  a  level  section,  as  S  is  to  S'  —  8,  and  (4)  shows  that  11'  R 
is  to  CK-as  Sis  to  S'. 

It  is  evident  that  these  formulas  apply  directly  to  the  lower 
side  of  an  embankment,  which  Fig.  147  will  represent  by  turning 
it  over. 

Example.—  Let  the  roadbed  be  20  feet  wide,  center  cut  10  feet, 
slope  of  the  ground  4  to  1,  or  8'  =  4,  and  side  slope  1  to  1,  or 
8  —  1.  Find  the  distance  out  to  H. 


™    i  ^       4(10 

We  have  CK  = 


4-1 


A!so, 


GIF  =  10  -f  10  =  20. 
On  the  lower  side  of  the  cut  we  Lave 


P'l,       d,  GL 

PL=PL>     a"d    S=PL' 


CONSTRUCTION. 
Dividing,  we  have 

£  -  CL  -  d~di 

~8  ~  "dt  ~       di     ' 
or  dt=LPf=-  .......     (5) 


Subtracting  this  from  CP'  =  d,  we  have 

-  d8' 


Dividing  (5)  by  (6),  we  have 

LP'       8 


Dividing  (5)  by  CP'  =  d,  we  have 
LP1  S 


CP>  -  s1 4-  8 ^ 


These  equations  show  that  the  subtractive  distance  LP'  is  to 
the  distance  CP',  for  a  level  section,  as  8  is  to  S'  +  8,  and  LP' 
is  to  CL  as  /S  is  to  8'. 

It  is  evident  that  these  formulas  apply  at  once  to  the  upper  side 
of  a  fill. 

Example.  —  Let  &  —  20,  c  =  10,  Sf  —  4,  and  8=1. 

To  find  CL,  the  distance  of  P  from  the  center  C. 


We  have  gj  =  4<'°  +  "»  =  16. 

4-f-l 


Also, 


Let  Fig.  148  represent  a  side-hill  section. 


290 


FIELD-MANUAL   FOK    ENGINEERS. 


I.  To  find  by  trial  the  position  of  H.  Let  e  —  10.4  and  is  =  1. 
It  will  be  convenient  to  consider  the  height  of  the  roadbed  equal 
to  0. 

Distance  out  for  level  section  —  d  =  CH'  =  CP'  —  10 -f  10.4  =  20.4. 

Suppose  we  judge  the  ground  near  H'  to  be  5  feet  higher  than 
at  G,  or  15.4  feet.  The  distance  out  corresponding  to  that  height 
would  then  be  5  feet  beyond  H' ',  or 


10 


i5.4  =  25.4  feet. 


But  the  ground  5  feet  beyond  H'  is  still  higher  than  at  //',  re- 
quiring a  still  further  distance  out.  Let  us  test  it  at  e,  26.5  feet 
out.  Suppose  we  find  the  ground  6.6  feet  higher  than  at  C.  This 


JS, 


FIG.  148. 


requires  a  distance  out  of  20.4  +  6.6  =  27  feet,  or  0.5  of  a  foot  be- 
yond e.  But,  as  before,  at  27  feet  the  ground  is  a  little  higher 
than  at  26.5  feet,  so  we  will  try  it  at  77,  27.2  feet  out,  for  example. 

Suppose  we  find  this  point  0.2  feet  above  e,  or  6.8  above  G, 
This  requires  a  distance  out  of  27.0  -f  0.2  =  27.2.  Hence  the 
height  and  distance  out  correspond  and  are  therefore  correct. 

On  the  lower  side  of  a  fill  we  would,  for  reasons  already  given, 
proceed  in  precisely  the  same  way. 

II.  To  find  the  position  of  P. 

Let  us  suppose  the  ground  near  P1  to  be  6  feet  lower  than  at  0. 
This  would  require  a  distance  out  of  20.4  —  6  —  14.4  feet. 

But  6  feet  inside  of  P'  the  ground  is  higher  tlian  at  P  ,  and, 
according  to  the  supposition,  less  than  6  feet  below  C,  and  re- 
quires, therefore,  a  distance  out  greater  than  14.4. 

Suppose  we  find  the  ground  at  e',  17.4  feet  out,  4.8  feet  below  C', 


CONSTRUCTION".  291 

or  5.6  feet.  This  requires  a  distance  out  of  20.4  —  4.8  —  15.6 
feet,  or  10  -f-  5.6  =  15.6  feet. 

But  at  15.6  feet  out  the  ground  is  higher  than  at  c',  and  there- 
fore the  distance  out  must  be  greater  than  15.6  feet. 

Suppose  we  now  find  the  elevation  at  P,  16  feet  out,  to  be  0.4 
above  e',  or  6.0  feet.  This  height  and  the  distance  out  now  corre- 
spond and  are  correct. 

We  are  now  prepared  to  state  two  important  principles  which 
will  greatly  facilitate  the  process  of  cross-sectioning  : 

I.  1  f,  on  the  upper  side  of  a  cut  or  lower  side  of  a  fill,  any  height 
found  calls  for  a  given  change,  e,  in  the  distance  out,  either  inward 
or  outward,  the  real  change  required  is  always  greater  than  e,  and 
the  excess  increases  with  the  lateral  steepness  of  the  ground. 

Thus,  referring  to  Fig.  149  and  the  preceding  notation,  ob- 
serving that  H"e'  is  the  additional  distance  out  (beyond  H')  cor- 
responding to  the  additional  height  11" H'  above  C,  we  have 

the  real  addi-  )  nv-  o> 

tional  dis-     [•  =  11' K  =  H"e  =  H"c'  -™  =  #"<•_—-        (9) 
tance  out      ) 

II.  If,  on  the  lower  side  of  a  cut  or  upper  side  of  a  fill,  the  height 
found  calls  for  a  given  change,   e,   in  the  distance  out,  either 
inward    or    outward,   the    real  change    required  is   always  less 
than  e,  and  the  deficiency  increases  with  the  lateral  steepness  of 
the  ground. 

We  have  as  above,  observing  that  P"d'  is  the  distance  to  come 
in  corresponding  to  the  fall  P' P"  below  0, 


l    ) 
e   U 
in  ) 


the  real 

distance  PK'  =  P'd  =  P"d'  X      =-  =P"d  -—-    (10) 

to  come  in 


The  author  has  used  these  principles  for  thirty  years,  but  has 
not  seen  them  stated. 

Example.  —  Let  ab  —  24,  c  =  10.5,  8  —  f,  to  find  the  position 
of  H.  (Fig.  149.)  CH'  =  12  -f  |(10.5)  =  27.8. 

Suppose  we  judge  the  ground  near  H'  (&iH")  to  be  6  feet  higher 
than  at  G.  Then  we  know  that  the  distance  out  is  greater  than 
27  g  -f-  9  —  36.8.  The  additional  distance  corresponding  to  H"H' 
i§  U"c',  and  eV  s=  36.8.  Suppose  we  try  it  at  e,  39.0  feet  out, 


O  O,  O 


FI ELD-M  A  X  I'  A  L    FO It   EKGIK E  ERS. 


and  find  e  8.6  feet  higher  than    C,  or  19.1  feet,  calling  for  a  dis- 
tance of  27.8  +  12.9  =  40.7  feet,  or  1.7  feet  beyond  e. 

We  will  therefore  try  it  at  H,  41.7  feet  out,  and  suppose  we 
find  the  ground  0.7  feet  higher  than  at  e.     This  would  call  for  an 


additional  foot  beyond  40.7,  or  41. 7  as  we  have  it.  The  cut  is 
therefore  19.1  -f  0.7  =  19.8,  and  distance  out  41.7  ft. 

To  find  the  position  of  P. 

We  have  CP'  =  27.8.  Suppose  we  judge  the  ground  near  P' 
(at  P")  to  be  6  feet  lower  than  at  C ';  then  we  know  that  P  is 
inward  from  P"  less  than  9  feet,  or  P"d'',  and  the  distance  out  is 
therefore  greater  than  27.8  —  9.0  =  18.8  feet  —  d'd". 

Suppose  we  try  it  at  ef,  22.0  feet  out,  and  find  e'  5.2  feet  lower 
than  C,  or  5.3  feet,  calling  for  a  distance  out  of  27.8  —  8.0  =  19.8 
feet,  or  2.2  inside  of  e'.  Suppose  we  come  in  1.2  feet  or  20.8  feet 
out  to  P,  and  find  it  0.6  feet  higher  than  at  e'  and  requiring  a  dis- 
tance out  of  19.8  -f  0.9  =  20.7.  As  this  is  but  0.1  inside  of  the 
point  of  observation,  we  can  probably  set  Pat  20.7  feet  out  with 
sufficient  accuracy. 

It  will  be  seen  that  the  first  point  of  observation  is  chosen  with 
reference  to  the  known  distance  CII'  for  a  level  section,  and  that 
subsequent  points  are  chosen  with  reference  to  the  figures  ob- 
tained at  the  last  point  observed. 

A  table  which  can  be  computed  in  a  few  minutes,  giving  dis- 
tances out  for  the  given  side  slope  corresponding  to  different 
heights,  will  greatly  facilitate  the  work  and  likewise  conduce  to 
accuracy. 

If  the  ground  is  irregular  laterally,  the  heights  must  be  found 
at  all  points  along  the  cross-section  where  the  slope  changes,  and 
these  heights,  as  well  as  their  distances  out,  must  be  carefully 
recorded. 

When  the  surface  of  the  ground  intersects  the  roadbed  as 
shown  in  Fig.  150,  we  have  what  is  called  side-hill  work. 


CONSTRUCTION.  293 

Let  POGH  represent  the  surface  of  the  ground.     H  is  found 
f.s  already  explained,  and  0  is  found  by  simply  finding  a  point  on 


FIG.  150. 


HC  prolonged,  whose  height  is  equal  to  the  height  of  the  roadbed 


or  aero.     To  find  P  we  have 


in  a 


=  8,  and  we  must  find  a  point 


Pon  CO  prolonged  that  will  satisfy  this  equation.  To  do  this  it  is 
only  necessary  to  observe  that  ma  increases  faster  than  mP ;  and 
ma 


hence  to  increase  - 


mP 


we  must  move  outward,  and  to  decrease 


the  same  we  must  move  inward. 

When  two  materials  are  found  in  the  same  section,  as  rock 
overlaid  with  earth,  it  is  necescary  to  give  each  material  its 
proper  slope. 

Since  the  upper  layrr  is  usually  of  varying  thickness,  it  will  be 
necessary  to  excavate  a  trench  along  the  cross-section  PCH  in 


FIG.  151. 

order  to  expose  the  rock,    so  as   to   set   the  stakes  PI   and 
Then  Pand  //can  be  located  as  usuaj. 


294  FIELD-MANUAL    FOR   ENGINEERS. 

It  is  easily  shown,  however,  that  when  the  layer  of  earth  is 
uniform 


+  f>  +  .,     .    .    (11) 


In  these  equations  S  =  side  slope  of  the  rock,  8t  =  that  of 
the  earth,  and  S'  =  the  slope  of  the  ground  ;  also  &  =  width  of 
the  roadbed,  c  =  center  depth  of  rock,  and  d  =  that  of  the  earth. 

BORROW-PITS.  —  When  the  cuts  are  insufficient  to  make  the 
fills,  or  are  too  far  away,  material  taken  from  borrow-pits  is 
used.  These  should  be  carefully  staked  out  by  the  engineer 
so  that  their  contents  can  be  calculated  when  completed. 

It  is  usual  to  lay  out  the  area  to  be  used  in  squares  or 
rectangles,  extending  one  or  both  sets  of  lines,  so  that  when 
the  excavation  is  completed  the  lines  can  be  readily  reproduced 
in  the  bottom  of  the  borrow-pit,  and  heights  taken  there  under 
the  original  heights  taken  upon  the  surface,  thus  giving  the 
depths  of  the  excavation. 

If  the  sides  of  the  rectangles  in  one  direction  are  10  feet 
or  some  multiple  of  10  feet,  and  in  the  other  direction  27  feet 
or  some  multiple  of  27  feet,  the  computation  is  very  much 
facilitated. 

Borrow-pits  should  be  regularly  excavated  so  that  they  can 
be  easily  computed  and  will  not  present  an  unsightly  appear- 
ance when  abandoned. 

Material,  when  suitable,  may  be  obtained  by  widening  the 
cuts,  provided  the  fills  are  accessible  and  not  too  far  away. 

Any  surplus  material  in  the  cuts  should  be  used  in  widening 
the  adjacent  fills  if  possible,  otherwise  it  should  be  deposited 
where  the  engineer  directs,  and  is  said  to  be  wasted.  It  should 
in  no  case  be  deposited  on  the  upper  side  of  a  cut,  unless  \veli 
removed  from  the  edge,  for  otherwise  it  would  greatly  interfere 
with  the  surface  ditches,  which  should  always  extend  along 
the  upper  side  of  a  cut,  a  few  feet  from  the  edge,  to  prevent 
the  surface  water  from  pouring  into  them. 

SHRINKAGE.  —  In  estimating  the  volumes  of  cuts  <m<!  fills  10 
be  made  from  them  we  have  to  bear  in  mind  that  earths  are 


CONSTRUCTION.  295 

more  compact  in  fills  than  in  cuts,  but  rock  is  less  so.     The 
shrinkage  in  the  bank  is  about  as  follows  for  different  materials: 

Gravel  and  gravelly  earth 8  per  cent. 

Gravel  and   sand 9    "       " 

Clay  and  clayey  earth 10    "       " 

Loam  and  light  sandy  soils 12    "       " 

Loose  vegetable  soils 15    "       " 

Puddled  clay 25    "       " 

Much,  however,  depends  upon  the  condition  of  the  material 
when  handled,  and  the  mechanical  appliances  used. 

Embankments  made  with  wheelbarrows  are  very  loose  and 
will  sin-ink  15  per  cent  or  more  if  of  a  clayey  or  gravelly 
nature,  and  perhaps  20  or  even  25  per  cent  if  loamy  or  of  loose 
vegetable  soil. 

Embankments  formed  with  wagons  do  not  shrink  so  much, 
and  those  put  up  by  means  of  drag-scrapers  or  wheel-scrapers 
shrink  least  of  all. 

In  view  of  shrinkage  engineers  should  have  embankments 
built  higher  than  they  are  desired  to  be  after  settling,  and 
should  have  the  stakes  set  accordingly. 

Thus  if  it  is  desired  to  have  an  embankment  10  feet  high 
after  settling,  it  should  be  cross- sectioned  and  built  11  feet  high, 
say. 

Some  engineers  have  the  stakes  marked  10  feet,  say,  and 
direct  the  contractor  to  put  in  an  extra  foot  of  earth,  but  since 
the  volumes  of  embankments  increase  more  rapidly  than  their 
heights  this- is  not  advisable. 

RETRACING  THE  LINE — As  the  grading  is  nearing  completion 
points  are  established  on  the  roadbed  from  the  reference-points, 
and  the  line  is  retraced,  setting  substantial  stakes  of  conven- 
ient heights  100  or  200  feet  apart  on  tangents,  and  about  50 
feet,  more  or  less,  on  curves. 

The  chain  used  should  correspond  in  length  with  the  one 
used  in  measuring  the  line,  otherwise  the  result  would  be  as 
shown  in  Chapter  VI. 

The  earthwork  must  be  checked  as  to  widths,  depths,  etc., 
and  any  additional  work  to  be  done  should  be  plainly  indicated 
on  stakes  set  for  the  purpose. 

The  engineer  must  see  that  the  grade  is  properly  "  dressed," 
the  side  ditches  fully  opened,  so  as  to  secure  ''continuous" 
drainacrel  the  surface  ditches  dii.o"  and  iiTmhRt,nir»tp.d  ptc. 


296  FIELD-MANUAL   FOR   ENGINEERS. 

Tile  drains  should  be  laid  in  the  bottom  of  the  ditches  in 
all  cuts  except  those  especially  dry.  These  tiles  have  a  decided 
effect  in  keeping  the  earth  on  both  sides  of  the  ditches  dry, 
thus  protecting  the  roadbed,  preventing  the  earth  from  the  face 
of  the  cut  from  sliding  into  the  ditches,  and  decreasing  the 
expense  of  maintenance. 

TRACK-LAYING. — Before  the  track  is  laid  the  center  line  is 
retraced  and  short  stakes  are  set,  each  of  which  is  centered. 
These  stakes  may  be  spaced  at  200  feet  on  tangents,  50  feet  on 
ordinary  curves,  and  25  feet  on  very  sharp  curves. 

The  ties  are  then  spaced,  so  many  per  rail  length,  leaving, 
however,  the  transit  points  uncovered. 

The  ties  are  aligned  on  one  side  of  the  road,  and  it  they  are 
of  uniform  lengths  both  ends  will  be  aligned.  The  rails  are 
then  spiked  to  gauge,  the  first  spikes  being  driven  near  a  center 
stake,  the  center  mark  of  the  gauge-bar  being  kept  over  the 
center  on  the  stake. 

Upon  curves  the  rails,  as  a  rule,  must  be  bent  to  conform  to 
the  curve  before  being  laid,  but  this  is  not  necessary  for  very 
flat  curves,  and,  moreover,  a  track  of  light  rails,  say  50  pounds 
per  yard  or  less,  can  be  easily  sprung  into  any  desired  curve. 

All  the  ties  to  be  used  should  be  laid  before  the  rails  are 
placed  upon  them,  otherwise  the  rails  are  likely  to  be  bent 
before  the  track  is  surfaced. 

Owing  to  expansion  of  the  rails  by  heat  a  space  must  be 
left  at  the  rail- joints.  The  highest  temperature  of  a  rail  in  the 
sun  is  about  130°  Fahr. 

The  expansion  of  iron  or  steel  per  100°  is  .0007  per  foot. 
Therefore  when  30-foot  rails  are  laid  at  a  temperature  near  the 
freezing-point  (32°),  or  100°  below  the  maximum,  the  space 
allowed  must  be  at  least  .0007  X  30  =  -021  foot>  or  -252  indl< 
or  fully  one  quarter  of  an  inch. 

At  80°  Fahr.,  or  50°  below  the  maximum,  it  need  be  only 
half  as  much. 

The  space  required  is,  of  course,  proportional  to  the  length 
of  the  rail. 

The  engineer  should  provide,  or  see  that  the  contractor  pro- 
vides, wedges  suitable  to  different  temperatures  in  which  the 
rails  are  laid,  and  also  see  that  thev  are  used,  far  too  small  a 
space  would  result  in  the  rails  being  forced  up  by  expansion, 
and  a  space  too  large  would  result  in  a  rough  track. 


CONSTRUCTION.  297 

Where  sidings  are  required  the  necessary  frogs  and  switch- 
ties  should  be  provided  in  advance,  so  that  they  may  be  put 
in  place  as  the  main,  track  is  laid. 

Heavy  plank  for  road-crossings  should  be  laid  as  soon  as  the 
rails  are  spiked  so  that  the  highway  travel  may  not  be  inter- 
rupted. 

CULVERTS. — For  small  openings  piping  answers  an  excellent 
purpose.  The  ground  must  be  brought  to  grade  and  well 
tamped  so  as  to  form  a  firm  and  homogeneous  bed  for  the  pipe. 
If  the  natural  soil  is  unsuitable  to  form  a  firm  bed,  a  quantity 
of  clay  should  be  provided  for  it  and  tamped  as  above.  The 
til>l><  r  surface  of  the  pipe  should  be  at  least  2  feet  below  grade. 

It  is  generally  bad  practice  to  use  stone  in  any  way  about 
these  culverts,  since  it  destroys  the  homogeneity  of  the  struc- 
ture and  thus  invites  scouring  and  destruction  of  the  cul- 
vert. 

The  author  has  constructed  many  pipe  culverts  without  the 
use  of  a  stone,  and  without,  so  far  as  known  to  him,  the  loss 
of  a  single  culvert. 

Open  wooden  culverts  are  well  ad^ted  for  somewhat  larger 
openings  in  low  embankments.  Any  defects  in  them  are  easily 
seen,  and  they  are  easily  renewed  or  replaced  if  desired.  Cov- 
ered culverts  are  sometimes  used  in  higher  embankments,  the 
covers  being  square  timbers  12  or  more  inches  in  thickness. 

The  walls  of  the  culvert  should  extend  1%  or  2  feet  below 
the  surface  of  the  ground,  and  should  be  well  tied  into  the  bank 
by  timbers  embedded  for  that  purpose. 

The  bed  of  the  culvert  should  be  left  undisturbed,  the  open- 
ings should  be  as  nearly  of  a  uniform  cross-section  as  possible, 
having  a  moderate  uniform  slope,  and  their  approaches  should 
be  as  straight  and  uniform  in  slope  as  practicable. 

Where  these  essentials  are  fully  observed  and  the  culverts 
are  properly  constructed  there  is  no  danger  of  their  washing 
out.  Culverts  are  often  destroyed  in  consequence  of  being 
so  small  as  to  partly  block  up  the  stream,  and  thus  disturb  its 
easy  and  natural  flow,  causing  whirls,  eddies,  washing  of  banks, 
etc.,  and  finally  undermining  and  perhaps  washing  out  the 
culvert. 

It  is  not  enough  that  a  culvert  is  able  to  discharge  all  the 
water  that  approaches  it;  for  safety  it  must  do  so  without  too 
much  disturbing  the  natural  and  normal  flow  of  the  water. 


FIELD-MANUAL    FOR   ENGINEER!?. 


Culverts  of  less  than  8  or  10  feet  opening  should  not  be  con- 
structed in  any  case. 

For  the  foundation  of  large  culverts  piles  can  often  bo  used 
to  advantage.  Such  a  foundation  is  comparatively  inexpensive; 
it  will  last  indefinitely  and  cannot  be  washed  out.  Moreover, 
the  structure  itself  can  be  so  fastened  together  and  to  the 
piles  as  to  be  immovable. 

Where  good  stone  is  plentiful,  arched  masonry  culverts, 
though  costly,  are  in  some  respects  the  best  of  all,  since  if 
properly  built  they  are  very  durable  and  therefore  not  likely 
to  fail  and  cause  disaster. 

The  location  of  a  culvert  depends  somewhat  upon  the  con- 
formation of  the  ground,  it  being  necessary  sometimes  (but 
should  be  avoided  when  practicable)  to  set  the  culvert  on  a 
"  skew,"  in  order  to  avoid  too  much  excavation  or  to  fit,  to  the 
best  advantage,  the  thread  of  the  stream. 

A  stake  is  set  at  each  corner  of  the  area  to  be  occupied  by 
the  culvert,  and  the  cut  to  be  made  is  marked  upon  it.  The 
general  principles  of  locating  points  and  lines  under  various 
conditions  having  been  fully  explained  in  preceding  chapters, 
including  Chapter  IV,  it  is  only  necessary  to  apply  them  to 
any  and  every  case  that  may  arise. 

The  location  of  bridge  piers  on  a  curve  requires,  however, 
something  of  a  compromise;   and  what  that  is,  and  a  handy 
way  of  doing  it,  will  therefore  be  explained  in  this,  connection. 
Let  CD,  Fig.  152,  represent  the  curve  between  two  adjacent 
piers.    Draw  the  chord  CD. 
The  middle  ordinate 

MN  =  CE  vers  CEM . 
Take   CA  =  DB  =  y2MN.      A   and 
B  are  the  centers  of  the  piers. 

We  observe  that  AB  is  on  a  line 
half-way  between  the  chord  CD  and 
tangent  to  the  curve  at  .!/. 

Proceed  in  the  same  way  for  other 
piers. 

TUNNELS, — Tunnels,  when  possible, 
should  be  on  a  tangent  throughout, 
so  as  to  be  easily  laid  out,  and  to 
freely  admit  the  light. 
The  location  of  Tunnels,  like  that  of  Bridges  and  other  struc- 


CONSTRUCTION'.  209 

tures  should  of  course  be  such  as  to  render  the  total  cost  a 
minimum  for  a  given  service  rendered. 

The  material  to  be  encountered  may  in  some  cases  be  deter- 
mined with  tolerable  accuracy  by  a  study  of  the  geology  of  the 
adjacent  region,  but  for  more  accurate  information  it  is  neces- 
sary to  resort  to  borings. 

The  alignment  of  a  tunnel  is  more  or  less  elaborate  according 
to  its  length  and  surroundings,  and  great  care  should  be  be- 
stowed upon  it.  Indeed  in  all  cases  where  errors  v/ould  be 
more  or  less  costly  every  effort  should  be  made  to  avoid  them 
by  using  the  best  tapes  and  rods  and  instruments,  in  perfect 
adustment,  and  by  repeating,  several  times,  all  observations 
with  transit,  level,  etc.,  and  all  computations.  It  is  of  the 
greatest  importance  to  have  the  transit  revolve  in  a  vertical 
plane,  and  to  secure  this  a  sensitive  bubble-tube  should  be 
attached  to  the  horizontal  axis  of  the  telescope. 

It  will  usually  be  necessary  to  find  the  distance  through  the 
tunnel  by  triangulation. 

By  triangulation  also,  and  by  direct  surveying,  the  highest 
point  and  other  high  points  on  the  line  of  the  tunnel  are  found, 
as  well  as  other  points  on  the  prolongation  of  the  line  in  both 
directions.  These  points  determine  the  line,  and  stations  are 
established  at  them,  by  means  of  which  any  desired  point  may 
be  located  and  the  work  easily  laid  out. 

As  the  excavation  of  the  tunnel  proceeds  successive  points 
on  the  center  line  are  determined,  usually  on  the  roof,  from 
which  plumb-lines  are  suspended,  thus  plainly  and  constantly 
defining  the  line. 

Whether  tunnels  are  on  tangents  or  curves  the  surveying 
operations  connected  with  them  are  simple  and  have  been  fully 
explained. 

As  already  stated,  great  care,  repetitions,  and  numerous  and 
varied  checks  on  the  work  are  important. 

For  detailed  and  full  information  regarding  tunnels  the 
reader  is  referred  to  Sims'  and  Drinker's  books  on  the  subject, 
and  to  the  current  engineering  journals. 


CHAPTER  XIII. 

EXPLANATION  OF  TABLES-  AND  MISCELLANEOUS  TOPICS. 

Table  I.    The  radius  of  a  1°  curve  =  ^^-A^. 


1  80  V  1  00  v  60 
The  radius  of  a  1'  curve  =          A          A        =  343774.677078. 


To  find  the  radius  of  any  curve  divide  the  ladius  of  a  1'  curve 
by  the  number  of  minutes  in  the  degree  of  the  given  curve. 

Table  II  contains  tangent  offsets  for  all  arcs  up  to  25  feet,  50 
feet,  or  100  feet  for  curves  of  different  degrees. 

Tables  III,  III«,  and  lllb  contain  tangential  offsets,  middle  01  • 
dinates,  and  chords  for  arcs  of  100  feet,  differing  in  degree  by  10'. 

Tables  IV  and  V  contain  the  value  of  long  chords  and  middle 
ordinates  to  long  chords  for  curves  from  0°  to  20°,  differing  by  10  . 
These  tables  are  very  useful  for  passing  obstacles,  for  finding  tho 
middle  point  of  any  arc,  for  laying  out  curves,  etc.,  etc. 

Table  VI  contains  the  various  elements  of  frogs  and  turnout 
curves. 

Table  VII  contains  the  true  values  of  the  tangents  and  externa  '•« 
of  a  V  curve.  To  find  these  functions  for  any  other  curve  divide 
the  tabular  numbers  corresponding  to  the  given  intersection  anglu 
by  the  number  of  minutes  in  the  degree  of  the  curve. 

Thus  for  a  2°  14'  curve  and  V  =  28°  we  have 

tangent  =  85712.7  -*-  134  =  639.7, 
and  external  =  10524.2  -s-  134  =  78.5. 

Thus  the  true  result  is  given  by  a  single  operation,  whereas  tbw 
usual  tables  require  three  operations  to  find  an  approximat  « 
result. 


EXPLANATION"   OF   TABLES;   MISCELLANEOUS   TOPICS.    301 

Tables  VIII,  IX,  and  X  are  supposed  to  be  in  the  most  conven- 
ient form  for  use. 

Table  XI  contains  the  correction  e,  in  feet,  corresponding  to  any 
distance,  D,  due  to  curvature  and  refraction. 

Tables  XII,  XIII,  and  XIV  need  no  explanation. 

Tables  XV  and  XVI  contain,  respectively,  the  offsets  and  tan- 
gent distances  of  transition  curves,  in  terms  of  the  degree  of  the 
offset  curve,  and  the  length  of  the  offset  or  transition  curve. 

Thus  for  an  offset  curve  of  6°  and  transition  curve  200  feet  long 
the  offset  by  Table  XV  is  1.74.  Also,  by  Table  XVI, 

-  -  AO  =  .03,     or    AO  =  100  -  .03  =  99.97. 

Table  XVII  contains  deflection  angles  for  transition  curves,  for 
one  to  five  chords,  iu  terms  of  the  angle  for  the  first  chord.  Thus 
if  the  deflection  angle  for  the  first  chord  is  2f. 3,  then  the  angles  for 
the  second,  third,  fourth,  and  fifth  chords  are  9'. 2,  20'. 7,  36'. 8, 
and  57'.  5,  respectively. 

Table  XVIII  contains  the  volumes  oi  earth,  etc.,  for  different 
slopes  and  bases. 

Table  XIX  applies  to  all  bases  having  slopes  of  1  to  1  or  1^  to  1. 

Example. — Let  base  ab  (Fig.  135)  =  20,  slopes  1  to  1. 

Let  CD  =  e  =  6.4.     Then  DE  —  10,  and  CE  ~  6.4  -f  10  =  16.4. 

Now  for  16.4  the  table  gives,  volume  for  HKE  —  996 
and  for  10  the  table  gives,  volume  for  cibE        —  370 

Therefore  the  desired  volume  for  abllK  =  626 

The  subgrade  volume  abEJ  being  constant  for  any  given  base 
and  slope  can  be  taken  from  the  table  once  for  all. 

A  table  designed  specially  for  any  base  can  be  used  as  above 
;'or  any  other  base  having  the  same  slopes. 

The  remaining  tables  perhaps  require  no  explanation. 

MISCELLANEOUS. 

To  Gauge  a  Stream  Approximately. — Take  some  body,  a  partly 
filled  bottle,  for  example,  that  will  float  nearly  submerged,  and 
allow  it  to  do  so  down  a  uniform  and  open  stretch  of  the  stream 
100  feet  in  length,  and  note  the  time  1  in  seconds. 


502  FIELt)-MAKUAL    FOR    ENGINEERS. 

Measure  tlie  cross-section  A  in  square  feet,  then  the  cubic  feet 
of  water  that  passes  per  minute  is 

5000  J. 


This  assumes  that  the  average  velocity  is  f  of  the  observed 
velocity.  (See  Bowser's  Hydromechanics,  p.  217.) 

Horse-power  of  falling  water  =  .Q0189QH,  in  which  Q  =  cubic 
feet  per  minute  falling  H  feet. 

Transverse  Strength  of  Rectangular  Beams.  —  Let  L  —  length 
in  feet,  b  =  the  breadth,  and  d  =  depth  in  inches  ;  w  =  load  at 
center  in  pounds,  and  R  =  modulus  of  rupture.  Then 

Kbd* 

w  =  — 


1SL  ' 

R  is  taken  from  1000  to  15,000  pounds  for  wood,  and  from  10,000 
to  15,000  pounds  for  wrought  iron. 

If  the  load  is  distributed,  it  will  carry  twice  the  above  amount. 
This  is  useful  during  construction  in  deciding  whether  or  not 
working  trains  can  safely  pass  over  unfinished  bridges,  culverts, 
etc. 

JSafe  Bearing-power  of  Piles. — Let  w  =  weight  in  pounds  of  the 
hammer  used  in  driving  a  pile,  h  —  the  fall  in  feet,  8  =  the 
penetration  in  inches,  and  R  =  the  bearing-power  sought.  Then, 
assuming  a  factor  of  safety  of  6, 


£+r 

This  is  the  formula  of  the  late  Mr.  A.  M.  Wellington,  formerly 
editor  of  the  Engineering  News.  It  has  the  merit  of  being  as 
trustworthy  as  any,  combined  with  grea*  simplicity. 


363 


TABLES. 


304      TABLE  I  —RADII.  CHORDS,  OFFSETS,  AND  ORDINATES. 


Degree 
D. 

Radius 
R. 

Chord 
Sta. 

Tang. 
Off. 
t. 

Mid. 
Ord. 
m. 

Degree 
D. 

Radius 
B. 

Chord 
1 
Sta. 

Tang 
Off. 
t. 

Mid. 
Ord. 
in. 

0' 

No  rad. 

100.00 

1°   0' 

5729.58 

.873 

.218 

1 

343775. 

.015 

.004 

1 

5635.65 

.887 

.2-22 

a 

171887. 

.029 

.007 

2 

5544.75 

.90:. 

.828 

JJ 

11459;!. 

.044 

.011 

3 

5456.74 

.916 

.  xixJ'J 

4 

85943.7 

.058 

.015 

4 

5371.48 

.931 

.233 

5 

68754.9 

.073 

.018 

5 

5288.84 

.945 

.236 

(i 

57295.8 

.087 

.022 

6 

5208.71 

99.998 

.900 

.240 

7 

49110.7 

.102 

.025 

7 

5130.97 

.974 

.244 

8 

42971.8 

.116 

.029 

8 

5055.51 

.989 

.247 

9 

38197.2 

.131 

.033 

9 

4982.24 

1.004 

.251 

10 

34377.5 

.145 

.036 

10 

4911.07 

1.018 

.255 

11 

31252.2 

.160 

.040 

11 

4841  .90 

1.033 

.258 

12 

28647.9 

.175 

.044 

12 

4774.05 

.047 

.2(52 

13 

26444.2 

.189 

.'047 

13 

4709.24 

.060 

.265 

14 

24555.3 

.204 

.051 

14 

4645.60 

.076 

.269 

15 

22918.3 

.218 

.055 

15 

4583.66 

.091 

.273 

16 

21485.9 

.'.'33 

.058 

16 

4523.35 

.105 

.276 

17 

202-22.0 

.247 

.062 

17 

4464.61 

.120 

.280 

18 

19098.6 

.262 

.065 

18 

4407.37 

.134 

.284 

19 

18093.4 

.276 

.0(59 

19 

4351.58 

.149 

.287 

20 

17188.7 

.291 

.073 

20 

4297.18 

.164 

.291 

21 

10370.2 

.305 

.0;.; 

21 

4244.13 

.178 

.295 

22 

15626.1 

.320 

.080 

22 

410-2.37 

.193 

MB 

23 

14946.7 

.33"^ 

.<SI 

23 

4141.  S(i 

.207 

.  302 

2t 

14323.9 

.349 

.(-87 

24 

4092.56 

.22^ 

.806 

25 

13751.0 

.864 

.091 

25 

4044.41 

99.997 

.236 

.309 

'38 

13222.1 

.378 

.OP5 

26 

3997.38 

.251 

.313 

2; 

12732.4 

.393 

098 

27 

3951.43 

.265 

.316 

28 

12277.7 

.407 

.10-2 

28 

3906.53 

.280 

.320 

29 

11854.3 

.422 

.105 

29 

3862.64 

.294 

.324 

30 

11459.2 

.436 

.109 

30 

3819.72 

.309 

.3-27 

31 

11089.5 

.451 

.113 

31 

3777.74 

.323 

.331 

32 

10743.0 

.465 

.116 

32 

3736.68 

.338 

.335 

33 

10417.4 

.480 

.120 

33 

3696.50 

.353 

.338 

34 

10111.0 

.495 

.124 

34 

3657.18 

.367 

.Mi 

35 

9822.13 

.509 

.127 

35 

3618.68 

.382 

.345 

36 

9549.30 

.524 

.131 

36 

3580.99 

.396 

.34!) 

87 

9291.21 

.538 

.135 

37 

3544.07 

.411 

.353 

38 

9046  71 

99.999 

.553 

.138 

38 

3507.90 

.425 

.356 

39 

8S14.74 

.067 

.142 

39 

3472.47 

.440 

.360 

40 

8594.37 

.582 

.145 

40 

3437.75 

99.996 

.454 

.364 

41 

83*4.75 

.596 

.149 

41 

3403  .71 

.469 

.367 

42 

8185.11 

.011 

.153 

42 

3370.34 

.483 

.371 

43 

7994.76 

.625 

.156 

43 

3337.62 

.498 

.375 

44 

7813.06 

.640 

.160 

44 

3305.53 

.513 

.378 

4i 

7639.44 

.654 

.164 

45 

3274.04 

.5-27 

.382 

4G 

7473.36 

.669 

.167 

46 

3243.16 

.542 

.385 

47 

7314.35 

.684 

.171 

47 

3212.85 

.556 

.389 

4H 

7161.97 

.698 

.175 

48 

3183.10 

.571 

.393 

49 

7015.81 

.713 

.178 

49 

3153.90 

.585 

.:•;<)(> 

50 

6875.49 

.727 

.182 

50 

3125.22 

.600 

.400 

51 

6740.68 

.742 

.185 

51 

3097.07 

.614 

.404 

52 

6611.05 

.756 

.189 

52 

3069.42 

.629 

.407 

53 

6486.31 

.771 

.193 

53 

3042.25 

99.995 

.643 

.411 

54 

6366.20 

.785 

.196 

54 

3015.57 

.658 

.415 

55 

6250.45 

.800 

.200 

55 

2989.35 

.672 

.418 

56 

6138.83 

.814 

.204 

56 

2963.57 

.687 

.42-2 

57 

6031.13 

.829 

.207 

57 

2938.25 

.  702 

.425 

58 

59-27.15 

.844 

.211 

58 

2913.34 

716 

.429 

59 

5826.69 

.858 

.215 

59 

2888.86 

.731 

.433 

CO 

5729.58 

.673 

.218 

60 

2864.79 

745 

.436 

1 

TABLE  I.-RAD1I,  CHORDS,  OFFSETS,  AND  ORDINATES.      305 


;  Degree 
D. 

Radius 
K. 

Chord 
1 
Sta. 

T<SSK- 

t. 

Mid. 
Orel. 

m. 

Deere. 

Radius 
R. 

Chord 
1 
Sta. 

To1£ 

t. 

31  id. 
Old. 
tit. 

'  2°  00' 

281)4.79 

.745 

.436 

3°  00' 

1909.86 

2.617 

.054 

1 

2841.11 

.760 

.440 

1 

1899.31 

99.988 

2.632 

.658 

2 

2817.83 

.774 

.444 

2 

1888.87 

2.646 

.662 

3 

2794.92 

.789 

.447 

3 

1878.55 

2.  (561 

.665 

4 

2772  38 

.803 

.451 

4 

1868.34 

2.676 

.669 

5 

x!7:>0.2( 

99.994 

.818 

.454 

5 

1858.24 

2.690 

.673 

6 

2728-37 

.83-.' 

.457 

6 

1848.25 

2.705 

.676 

7 

2706.89 

.847 

.462 

7 

1838.37 

2.719 

.680 

8 

2685.74 

.861 

.465 

8 

1828.59 

2.734 

.684 

9 

2664.92 

.87o 

.468 

9 

1818.91 

99.987 

2.74H 

.687 

10 

2044.42 

.891 

.473 

10 

1809.34 

2.763 

.691 

11 

2624.23 

.905 

.476 

11 

1799.87 

2.777 

.694 

12 

2604.35 

.920 

.480 

12 

1790.49 

2.792 

.698 

13 

2584.77 

.934 

.484 

13 

1781.22 

2.806 

.702 

14 

2565.48 

.9-19 

.487 

14 

772.03 

2.821 

.705 

15 

2546.48 

.983 

.491 

15 

762.95 

2.835 

.709 

16 

2527  75 

99.993 

1.978 

.494 

16 

753.95 

99.986 

2.850 

.713 

17 

2509  .'30 

1.992 

.498 

17 

745.05 

2.864 

.716 

18 

2491.12 

2.007 

.502 

18 

736.24 

2.879 

.720 

19 

2473.20 

2.021 

.505 

19 

727.51 

2.894 

.724 

20 

2455.53 

2.036 

.509 

20 

718.87 

2.908 

727 

21 

2438.12 

2.050 

.513 

21 

1710.32 

2.923 

.731 

2-2 

2420.95 

2.065 

.516 

22 

1701.85 

2.937 

.734 

23 

2404.0'J 

2.080 

.520 

23 

1693.47 

99.985 

2.952 

.738 

24 

2387.32 

2.094 

.524 

24 

1685.17 

2.96'i 

.743 

.  ^5 

2370.86 

2.109 

.527 

25 

1676.95 

2.981 

.745 

26 

2354.62 

99.992 

2.123 

.531 

26 

1668  81 

2.995 

.749 

27 

2338.60 

2.138 

.534 

27 

1660.75 

3.010 

.753 

28 

2322.80 

2.152 

.538 

28 

1652.76 

3.024 

.756 

29 

2307.21 

2.167 

.542 

29 

1644.85 

3.039 

.760 

30 

2291.83 

2.181 

.545 

30 

1637.02 

99.984 

3.053 

.764 

31 

2276.65 

2.196 

.549 

31 

1629.26 

3.068 

.767 

32 

2261.68 

2.210 

.553 

32 

1621.58 

3.082 

.771 

33 

2246.89 

2.225 

.556 

33 

1613.97 

3.097 

.774 

34 

2232.30 

2  239 

.560 

34 

1606.42 

3.112 

778 

35 

2217  90 

2.254 

.564 

35 

1598.95 

3.126 

.783 

36 

2203.68 

99.991 

2.269 

.567 

36 

1591.55 

3.141 

.785 

37 

2189.65 

2.283 

.571 

37 

1584.22 

99.983 

3.155 

789 

38 

2175.79 

2.298 

.574 

38 

1576.95 

3.170 

.793 

39 

2162.10 

2.312 

.578 

39 

1569.75 

3.184 

.796 

40 

2148.59 

2.327 

.582 

40 

1562.61 

3.199 

.800 

41 

2135.25 

2.341 

.585 

41 

1555.54 

3.213 

.804 

4-2 

2122.07 

2.356 

.589 

42 

1548.53 

3.2^8 

.807 

43 

2109.05 

2.370 

.593 

43 

1541.59 

99.982 

3.242 

.811 

44 

2096.19 

2.385 

.596 

44 

1534.71 

3.257 

.814 

45 

2083.48 

99.990 

2.399 

.600 

45 

1527.89 

3.  -271 

.cSKS 

46 

2070.93 

2.414 

.604 

46 

1521.13 

3.28(1 

.822 

47 

2058.53 

2.428 

.607 

47 

1514.43 

3.300 

.825 

48 

2046.28 

2.443 

.611 

48 

1507.78 

3.315 

.8.9 

49 

2034.17 

2.  458 

.614 

49 

1501.20 

3.329 

.P33 

50 

2022.20 

2  472 

.618 

50 

1494.67 

99.981 

3.344 

.830 

51 

2010.38 

2.487 

.622 

51 

1488.20 

3.358 

.840 

52 

1998.69 

2.501 

.625 

52 

1481.79 

3.373 

.843 

53 

1987.14 

99.989 

2.516 

.629 

53 

1475.43 

3.388 

.847 

54 

1975.72 

2.530 

.633 

54 

1469.12 

3.402 

851 

55 

1964.43 

2.545 

.636 

55 

1462.87 

3.417 

.854 

56 

1953.27 

2.559 

.640 

56 

1456.67 

99.980 

3.43i 

.858 

57 

1942.23 

2.574 

.644 

57 

1450.53 

3  446 

.8f2 

58 

1931.32 

2.588 

.647 

58 

1444.  41 

3.460 

.865 

59 

1920.53 

2.603 

.651 

59 

1438.39 

3.475 

869 

60 

1909.86 

2.617 

.654 

60 

1432.39 

1 

3.489 

'.873 

306  TABLE  I. -RADII,  CHORDS,  OFFSETS,  AND  ORDINATES. 


Degree 
p. 

Radius 
K. 

Chord 
1 

Sta. 

Tang. 
Off. 
t. 

Mid. 
Ord. 
m. 

Degree 

Radius 
R. 

Chord 
1 
Sta. 

Tsr 

t. 

Mid. 
Ord. 
m. 

4°    0' 

1432.39 

3.489 

.873 

5°     0' 

1145.92 

4.361 

.091 

1 

1420.45 

3.504 

.876 

1 

1142.11 

4.375 

.094 

2 

1420.50 

99.979 

3.518 

.880 

2 

1138.33 

4.390 

.098 

3 

1414  71 

3.533 

.883 

3 

1134.57 

4.404 

.102 

4 

1408.91 

3.547 

.887 

4 

1130.84 

99.967 

4.419 

.105 

5 

1103.10 

3.562 

.891' 

5 

1127.13 

4.433 

1.109 

6 

1397.40 

3.570 

.894 

6 

1123.45 

4.448 

1.112 

7 

1391.80 

99.978 

3.591 

.898 

7 

1119.79 

4.462 

1.116 

8 

1386.19 

3.605 

.902 

8 

1116.15 

4.477 

1.120 

9 

1380.62 

3.020 

.905 

9 

1112.54 

99.960 

4.491 

1.123 

10 

isrs.io 

3.635 

.909 

10 

1108.95 

4.506 

1.127 

11 

1339.62 

3.649 

.913 

11 

1105.38 

4.520 

.131 

IsJ 

1304.19 

3.664 

.916 

12 

1101.84 

4.535 

134 

13 

13:8.79 

99.977 

3.678 

.920 

13 

1098.32 

99.965 

4.549 

.138 

14 

1353.44 

3.693 

.923 

14 

1094.82 

.564 

.142 

15 

1348.14 

3  707 

.927 

15 

1091.35 

.578 

.145 

16 

1342.87 

3.722 

.931 

16 

1087.89 

.593 

.149 

17 

1337.64 

3.736 

.934 

17 

1084.46 

.607 

.152 

18 

1332.46 

3.751 

.938 

38 

1081.05 

09.964 

.  622 

.156 

19 

1327.32 

99.976 

3.765 

.942 

19 

1077.00 

.636 

.160 

20 

1322.21 

3.780 

.945 

20 

1074.30 

4.651 

.163 

521 

1317.14 

3.794 

.949 

21 

1070.95 

4.R65 

.167 

22 

1312.12 

3.809 

.953 

22 

1067.02 

99.963 

4.680 

.171 

23 

1307.13 

3.823 

.956 

23 

1064.32 

4.694 

.174 

24 

1302.18 

99.975 

3.838 

.960 

24 

1061.03 

4.709 

.178 

25 

1297.26 

3.852 

.963 

25 

1057.77 

4.723 

.182 

26 

1292.39 

3.867 

.967 

26 

1054.52 

4.738 

'   .185 

27 

1287.55 

3.881 

.971! 

27 

1051.30 

99.962 

4.752 

.189 

28 

1282.74 

3.890 

.974' 

28 

1048.09 

4.767 

.192 

29 

1277.97 

99.97-) 

3.910 

.978 

29 

1044.91 

4.781 

.196 

30 

1273.24 

3.925 

.982' 

30 

1041.74 

4.796 

.200 

31 

1268.51 

3.939 

.985 

31 

1038.59 

99.961 

4.810 

.203 

88 

1263.88 

3.954 

.989! 

32 

1035.47 

4.825 

.207 

33 

1259.25 

3.969 

.993 

33 

1032.36 

4.840 

.211 

34 

1254.65 

3.983 

.996 

34 

1029.27 

4.854 

.214 

35 

1250.09 

90.973 

3.998 

1.000 

35 

1026.19 

99.960 

4.869 

.218 

36 

1245.56 

4.012 

.003 

36 

1023  14 

4.883 

.221 

37 

1241.06 

4.027 

.007 

37 

1020.10 

4.898 

.225 

38 

1236.60 

4.041 

.011 

38 

1017.08 

4.912 

.229 

39 

1232.17 

4.056 

.014 

39 

1014.08 

99.i>59 

4.927 

.232 

40 

1227.77 

99.972 

4.070 

.018 

40 

1011.10 

4.941 

.236 

41 

1223.40 

4.085 

.022 

41 

1008.14 

4.956 

.240 

42 

1219.06 

4.099 

.025 

42 

1005.19 

4.970 

.243 

43 

1214.75 

4.114 

.029 

43 

1002.26 

4.985 

.247 

44 

1210.47 

4.128 

.033 

44 

999.345 

99.958 

4.1)99 

.251 

45 

1206.23 

99.971 

4.143 

.036 

45 

996.448 

5.014 

.254 

46 

1202.01 

4.157 

.040 

46 

993.569 

5.028 

.258 

47 

1197.8,' 

4.172 

.043 

47 

990.7'05 

5.043 

.261 

48 

1193.66 

4  186 

.047 

48 

987.858 

99.957 

5.057 

.265 

49 

1  189.53 

4.201 

.051 

49 

985.028 

5  072 

.269 

50 

1185.43 

99.970 

4.215 

.054 

50 

982.213 

5.080 

.272 

51 

1181.36 

4.230 

.058 

51 

979.415 

5.101 

.276 

52 

1177.31 

4.244 

.062 

52 

176.633 

99.956 

5.115 

.280 

53 

1173.29 

4.259 

.005 

53 

973.866 

5.130 

.283 

54 

1109.30 

4.273 

.069 

54 

971.115 

5.144 

.287 

55 

1165.34 

99.069 

4.288 

.072 

55 

968.379 

5.159 

.291 

56 

1161.40 

4  302 

.076 

56 

965.659 

99.955 

5.173 

.294 

57 

1157.49 

4.317 

.080 

57 

962.954 

5.188 

.298 

58 

1153.61 

4.332 

.083 

58 

900.264 

5.20^ 

.301 

f>9 

1149.75 

99.9G8 

4.346 

.087 

59 

957.590 

5.217 

.305 

00 

1145.92 

4.301 

.091 

60 

954.930 

99.954 

5.231 

.3091 

TABLE  I.— RADII,  CHORDS,  OFFSETS,  AND  ORDINATES. 


Degree 
D. 

Radius 
R. 

Chord 
1 
Sta. 

ToT- 
t. 

Mid. 
Ord. 
m. 

Degree 
D. 

Radius 
R. 

Chord 
1 
Sta. 

Tang. 
Off 
t. 

Mid. 
Ord. 
m. 

6°  00' 

954.930 

5.231 

.309 

7°  00' 

818.511 

6.101 

.527 

1 

952.284 

5.246 

.312 

1 

816.567 

6.116 

.530 

94:t.  654 

5.260 

.316 

2 

814.632 

99.937 

6.130 

.534 

a 

947.038 

5.275 

.320 

3 

812.706 

6.145 

.538 

4 

944.436 

99.953 

5.289 

.323 

4 

810.789 

6.159 

.541 

5 

941.848 

5.304 

.327 

5 

808.882 

99.936 

6.174 

.545 

6 

939.275 

5.318 

.330 

6 

806.983 

6.188 

.548 

7 

936.716 

5.333 

.334 

7 

805.093 

6.202 

.552 

8 

934.170 

99.952 

5.347 

.338 

8 

803.212 

99.935 

6.217 

.556 

9 

931  .639 

5.362 

.341 

9 

801  .340 

6.231 

.559 

10 

929.121 

5.376 

.345 

10 

793.476 

6.246 

.503 

11 

926.616 

99.951 

5.391 

.349 

11 

797.621 

6.260 

.567 

19 

924.125 

5.405 

.352 

12 

795.775 

99.934 

6.275 

.570 

18 

921.648 

5.420 

.356! 

13 

793.937 

6.289 

.574 

14 

919.184 

5.434 

.360: 

14 

792.108 

6.304 

.578 

15 

916.732 

99.950 

5.449 

3S3 

15 

790.287 

99.933 

6.318 

.581 

16 

914.294 

5.463 

.367 

16 

788.474 

6.333 

.585 

17 

911.869 

5.478 

.370 

17 

786.670 

6.347 

.588 

18 

909.457 

5.492 

.374 

18 

784.874 

99.932 

6.  302 

.592 

19 

907.057 

99.949 

5.507 

.378 

19 

783.058 

6.376 

.590 

20 

904.070 

5.521 

.381 

20 

781.306 

0.391 

.599 

21 

902.296 

5.536 

.3851 

21 

779.534 

99.931 

6.405 

.603 

22 

899.934 

5.550 

.389; 

22 

777.771 

6.420 

.607 

28 

897.584 

99.948 

5.565 

.392! 

23 

776.015 

6.434 

.610 

24 

895.247 

5.579 

.396' 

24 

774.267 

6.449 

614 

25 

892.921 

5.594 

.400i 

25 

772.527 

99.930 

6.463 

.618 

26 

890.608 

99.947 

5.608 

.403! 

26 

770.795 

6.478 

.621 

888.307 

5.623 

.407 

27 

769.071 

6.492 

.625 

28 

8S6.017 

5.637 

.410 

28 

767.354 

99.929 

6  507 

.628 

29 

S83.740 

5.652 

.414, 

29 

765.645 

6.521 

.632 

30 

881.474 

99.946 

5  666 

.418, 

30 

763.944 

6.536 

.036 

31 

879.219 

5.681 

.421! 

31 

762.250 

99.928 

6.550 

.639 

32 

876.976 

5.695 

.4251 

32 

760.  56S 

6.565 

.643 

?a 

874.745 

5.710 

.429 

33 

758.885 

6.579 

.047 

84 

872.525 

99.945 

5.724 

.432; 

34 

757.213 

99.927 

6.594 

.651 

as 

870.316 

5.739 

.436i 

35 

755.549 

6.  60S 

.054 

36 

808.118 

5.753 

.439 

36 

753.892 

6.623 

.657 

37 

865.931 

99.944 

5.768 

.443| 

37 

752.242 

99.926 

6.637 

.061 

38 

863.755 

5.782 

.4471 

38 

750.600 

6.651 

.605 

3!) 

861.591 

5.797 

.451 

39 

748.964 

6.060 

.668 

40 

859.437 

5.811 

.454! 

40 

747.336 

99.925 

6.080 

.67;: 

41 

857.293 

99.943 

5.826 

.458 

41 

745.715 

6.695 

.676 

42 

855.161 

5.840 

.461 

42 

744.101 

6.709 

.079 

43 

853.039 

5.855 

.465 

43 

742.494 

99.924 

6.724 

.683 

44 

850.927 

99.942 

5.869 

.469 

44 

740.894 

0.738 

.087 

45 

848.826 

5.884 

.472 

45 

739  .  SOO 

0.753 

.090 

46 

846.736 

5.898 

.476 

46 

737.714 

99.923 

0.707 

.694 

47 

844.655 

5.913 

.479 

47 

736.134 

6.782 

.097 

48 

842.585 

99.941 

5.927 

.483 

48 

734.561 

0.790 

.701 

49 

840.525 

5.942 

.487 

49 

732.997 

99.922 

6.811 

.705 

50 

838.475 

5.956 

.490 

50 

731.435 

0.825 

.708 

51 

836.435 

99.940 

5.971 

1.494 

51 

729.883 

6.840 

.712 

52 

834.405 

5.985 

1.498 

52 

728.336 

99.921 

6.851 

.716 

53 

832.384 

6.000 

1.501 

53 

726.796 

6.  869 

.719 

54 

830.374 

6.014 

1.505 

54 

725.263 

6.883 

.723 

55 

828.373 

99.939 

6.029 

1.509 

55 

723  730 

99.920 

6.898 

.726 

56 

826.381 

6.043 

.512 

56 

722.216 

6.912 

.730 

57 

824.400 

6  058 

.516 

57 

720.702 

0.927 

734 

58 

822.427 

99.938 

6.072 

.519 

58 

719.194 

99.919 

0.941 

.737 

59 

820.465 

6.087 

.523 

59 

717.692 

6.056 

.741 

60 

818.511 

6.101 

.527 

60 

716.197 

0.970 

.745 

308      TABLE  I.-RADII,  CHORDS,  OFFSETS,  AND  ORDINATES. 


Degree 
D. 

Radius 
R. 

Chord 
1 
Sta. 

Tang. 
Off 
t. 

Mid.  1 
Ord.  \ 

HI. 

Degree 
D. 

Radius 
R. 

Chord 

1 
Sta. 

Taug. 
Off. 
t. 

Mid. 
Ord. 

8°  00' 

716  197 

6.970 

T745J 

9°  00' 

630.620 

.838 

1.963 

1 

714.708 

99.918 

6.984 

.748 

1 

635.443 

.852 

1.966 

2 

713.2-25 

6.999 

.  4  52 

o 

634.271 

99.896 

.867 

1.970 

3 

711.749 

7.013 

.756 

3 

63  5.  103 

.881 

1  973 

4 

710.278 

99.917 

7.028 

.759' 

4 

631.939 

.890 

1  977 

5 

?08.814 

7.012 

.763 

5 

630.779 

99.895 

.910 

1.981| 

6 

707.355 

7.037 

.766! 

6 

629.624 

.9','t 

1.9841 

7 

705.903 

99.916 

7.071 

.770 

n 

028.473 

.939 

1.988 

8 

704.456 

7.086 

.774 

8 

627.326 

99.894 

.953 

1.992 

9 

703.016 

7.100 

9 

626.183 

.868 

1.995 

10 

701.581 

99.915 

7.115 

'.m 

10 

625.045 

99.893 

.982 

1.999 

11 

700.152 

7.129 

.785 

11 

623.910 

7.997 

2.002 

18 

098.729 

7.144 

.788 

12 

622.780 

8.011 

2.H06 

13 

697.312 

99.914 

7.158 

.792 

13 

621.654 

99.892 

8.026 

2.010 

14 

695.900 

7.173 

.795J 

14 

620.532 

8.040 

2.013 

15 

694.494 

7.187 

.799 

15 

619.414 

99.891 

8.055 

2.017 

16 

693.094 

99.913 

7.202 

.803, 

16 

61  S.  800 

8.069 

2.T21 

17 

691.700 

7.216 

.806 

17 

617.190 

8.084 

2.024 

18 

690.311 

7.230 

.810 

18 

616.084 

99.890 

8.098 

2.028 

19 

688.927 

99.912 

7.245 

.814' 

19 

614.982 

8.112 

2.031 

20 

687.541 

7.259 

.817 

20 

613.883 

99.889 

8.127 

2.035 

21 

686.177 

7.274 

.820 

21 

612.789 

8.141 

2.039 

22 

684.810 

99.911 

7.288 

.825 

22 

611.699 

8.156 

2.042 

23 

683.449 

7.303 

.828 

23 

610.612 

99.888 

8.170 

2.046 

24 

682.093 

99.910 

7.317 

.832 

24 

609.530 

8.185 

2.050 

25 

680.742 

7.3J-2 

.835 

25 

608.451 

99.887 

8.199 

2.053 

2G 

679.397 

7.346 

.839 

2d 

607.376 

8.214 

2.057 

27 

678.057 

99.909 

7.361 

.843 

27 

606.305 

8.228 

2.061 

28 

676.722 

7.375 

.846 

28 

605.237 

99.886 

8.243 

2.064 

29 

675.392 

7.390 

.850 

29 

604.173 

8.257 

2.068 

30 

G74.0H8 

93.908 

7.401 

.854 

30 

603.113 

99.885 

8.271 

2.071 

31 

672.749 

7.419 

.8)7 

31 

602.057 

8.286 

2.075 

«3 

071.430 

7.433 

.861 

32 

001.005 

K.300 

2.079 

33 

670.126 

99.907 

7.447 

.864 

33 

599.956 

99.884 

8.315 

2  082 

31 

60S.  822 

7.462 

.868 

34 

598.911 

8.3-29 

2.086 

35 

667.524 

7.476 

.872 

35 

597.8(i9 

99.883 

8.344 

2.01)0 

3G 

660.230 

99.906 

7.491 

.875 

36 

596.831 

8.358 

2  093 

37 

6(54.941 

7.505 

.879, 

37 

595.797 

8.372 

2.097 

88 

663.658 

99.905 

7.520 

.883' 

38 

594.760 

99.882 

8.387 

2.100 

39 

602.379 

7.534 

.886 

39 

593  739 

8.401 

2.104 

40 

661.105 

7.541) 

.890 

40 

592.71  5 

8.416 

2.108 

41 

659.836 

99.904 

7.563 

.894 

41 

591.695 

99.881 

8.430 

2.111 

42 

658.572 

7.578 

.897 

42 

590.678 

8.445 

2.115 

43 

057.313 

7.592 

.901 

43 

589.665 

99.880 

8.459 

2.119 

44 

656.059 

99.903 

7.607 

.904 

44 

588.u55 

8.474 

2.122 

45 

654.809 

7.621 

.908 

45 

587.649 

99.87'9 

8.488 

2.126 

46 

653-564 

99.902 

7.685 

.912 

46 

586.64ii 

8.502 

2.1  -21* 

47 

05J.324 

7.650 

.915 

47 

585.647 

8  517 

2.133 

48 

651  .089 

7.664 

.919 

48 

584.651 

99.878 

8.53! 

2.13; 

49 

649.858 

99.901 

7.679 

.923 

49 

583.658 

8.546 

2.140 

50 

648.631 

7.693 

.926 

50 

582.669 

99.877 

8.560 

2.144 

51 

647.410 

7.708 

.930 

51 

581.  683 

8.575 

2.118 

52 

646.193 

99.900 

7.722 

.933 

52 

580.700 

8.58!) 

2.151 

53 

644.980 

7  737 

.937 

53 

579.721 

99.876 

8.603 

2.155 

54 

643.773 

99.899 

7.751 

.941 

54 

578.745 

8.618 

2.159 

55 

642.569 

7.766 

.944 

55 

577.773 

99.875 

S.<532 

3.102 

56 

641.371 

7.780 

.948 

56 

576.803 

8.647 

2.166 

640.176 

99.898 

7.794 

.952 

57 

575.837 

99.874 

8.6til 

2.  l(i!> 

58 

638.986 

7.809 

.955 

|         58 

674.874 

8.676 

2.173 

59 

637.801 

7.623 

.959 

51) 

5  ;'.;.;>i  i 

8.  duo 

2.177 

GO 

636.620 

99.897 

7.838 

1.963 

60 

572.958 

99.873 

8.705 

2.  ISO 

TABLE  I.— RADII,   CHORDS,   OFFSETS,  AND  ORDINATES.     309 


Degree 

Radius 

Chord 
1 

Tang. 
Off. 

Mid. 
Orel. 

Degree 

Radius 

E> 

Chord 

1 

Tang. 
Off. 

Mid. 
Ord. 

D. 

R. 

Sta. 

t. 

me, 

D. 

£1. 

Sta. 

t. 

m. 

10°  OK 

572.958 

99.873 

8.705 

2.180 

20°  00' 

286.479 

99.493 

17.  -277 

4.352 

10 

563.5(55 

99.869 

8.849 

2.217 

10 

284.111 

99.485 

1  .418 

4.388 

20 

554.475 

99.864 

8.993 

2.253 

20 

281.783 

99.476 

1  .559 

4.424 

30 

5-15.674 

99.860 

9.137 

2.289 

30 

279.492 

99.467 

.700 

4.460 

40 

537.148 

99.856 

9.28-2 

2.325 

40 

277.238 

99.459 

1  .840 

4.497 

50 

528.884 

99.851 

9.426 

2.362 

50 

275.020 

99.450 

1  .981 

4.533 

11  00 

520.871 

99.847 

9.570 

2.398 

21  00 

272.837 

99.441 

18.122 

4.509 

10 

513.097 

99.842 

9.714 

2.434 

10 

270.689 

99.432 

18.262 

4.  605 

20 

505.551 

99.837 

9.858 

2.471 

20 

268.574 

99.423 

18.403 

4.641 

30 

498.224 

99.832 

10.002 

2.T07 

30 

266.49-2 

99.414 

18.543 

4.677 

40 

491.107 

99.827 

10.146 

2.543 

40 

264.442 

99.405 

18.68J 

4.713 

50 

484.190 

99.822 

10.290 

2.579 

50 

262.423 

99.396 

18.824 

4.749 

12  00 

477.465 

99.817 

10.434 

2.616 

22  00 

260.435 

99.3*7 

18.964 

4.785 

10 

470.924 

99.812 

10.578 

2.652 

10 

258.477 

99.378 

19.104 

4.821 

20 

464.560 

99.807 

10.721 

2.688 

20 

256.548 

99.36S 

19.244 

4.857 

30 

458.8158 

99.802 

10.865 

2.724 

30 

254.648 

99.359 

19.384 

4.8  3 

40 

-152.335 

99.797 

11.009 

2.761 

40 

252.775 

99.349 

19.524 

4.929 

ro 

446.461 

99.791 

11.152 

2.797 

50 

250.930 

99.340 

19.664 

4.965 

13  00 

440.737 

99.786 

11.296 

2.833 

23  00 

249.112 

99.330 

19.803 

5.001 

10 

435.158 

99.780 

11.440 

2.869 

10 

247.320 

99.3:0 

19.943 

5.037 

20 

429.718 

99.775 

11.583 

2.906 

20 

245.553 

99.310 

20.082 

5.073 

30 

424.413 

99.769 

11.727 

2.942 

30 

243.812 

90.301 

20.222 

5.100 

40 

419.237 

99.763 

11.870 

2.978 

40 

242.095 

99.291 

20.361 

5.145 

50 

414.186 

99.757 

12.013 

3.014 

50 

240.402 

99.281 

20.500 

5.181 

14  00 

409  256 

99.751 

12.157 

3.051 

24  00 

238.732 

99.271 

20.639 

5.217 

10 

404.441 

99.745 

12.300 

3.087 

10 

237.0815 

99.260 

20.779 

5.253 

20 

31(9.738 

99.739 

12.443 

3.123 

20 

235.462 

99.250 

20.918 

5.289 

30 

395.143 

99.733 

12.586 

3.159 

30 

233.860 

99.240 

21.056 

5.325 

40 

390.653 

99.727 

12.729 

3.195 

40 

232.280 

99.230 

•21.195 

5.361 

50 

386.264 

99.721 

12.872 

3.232 

50 

230.121 

99.219 

21.334 

5.391 

15  00 

381.972 

99.715 

13.015 

3.268 

25  00 

229.  IbS 

99.208 

21.473 

5.433 

10 

377.774 

99.108 

13.158 

3.304 

10 

227.665 

99.198 

21.611 

5.468 

20 

373.6t>8 

99.702 

13.301 

3.340 

20 

226.168 

99.187 

21.750 

5.504 

30 

369.650 

99.695 

13.444 

3.376 

30 

224.689 

99.177 

21.888 

5.540 

40 

365.718 

99.689 

13.587 

3.413 

40 

223.230 

09.1C6 

22.026 

5.516 

50 

361.868 

99.682 

13.729 

3.449 

50 

221.790 

99.155 

22.164 

5.612 

16  00 

358.099 

99.675 

13.872 

3.485 

26  00 

220.368 

99.144 

22.303 

5.648 

10 

354.407 

99.669 

14.015 

3.521 

10 

218.965 

99.133 

22.441 

5.684 

20 

3.50.790 

99.662 

14.157 

3.557 

20 

217.579 

99.122 

22.578 

5.720 

30 

347.247 

99.655 

14.300 

3.504 

30 

216.210 

99.111 

22  716 

5.756 

40 

343.775 

99.648 

14.442 

3.630 

40 

214.859 

99.100 

22.854 

5.792 

50 

340.371 

99.641 

14.584 

3.667 

50 

213.525 

99.089 

22.992 

5.827 

17  00 

337.034 

99.634 

14.727 

3.702 

27  00 

212.207 

99.077 

23.129 

5.863 

10 

333.762 

99.626 

14.869 

3.738 

10 

210.905 

99.066 

23.267 

5.  899 

20 

330.553 

99.619 

15.011 

3.774 

20 

209.619 

99.054 

23.404 

5.935 

30 

327.404 

99.612 

15.153 

3.810 

30 

208.348 

99.043 

23.541 

5.971 

40 

324.316 

99.604 

15.295 

3.847 

40 

207.093 

99.031 

23.678 

6.007 

50 

321.285 

99.597 

15.437 

3.883 

50 

205.853 

99.020 

23.815 

6.042 

18  00 

318.310 

99.589 

15.579 

3.919 

28  00 

204.628 

99.008 

23.952 

6.078 

10 

315.390 

99.582 

15.721 

3.955 

10 

203.417 

98.996 

24.089 

6.114 

20 

312.  52.  ' 

99.5^4 

15.863 

3.991 

20 

202.220 

98.984 

24*226 

6.150 

30 

309.707 

99.566 

16.005 

4.027 

30 

201.038 

98.972 

24.362 

6.186 

40 

306.942 

99.558 

16.146 

4.063 

40 

199.869 

98.060 

24.499 

6.222 

50 

304.225 

99.550 

16.288 

4.100 

50 

198.714 

98.948 

24  .  635 

6.257 

19  00 

801.557 

99.542 

16.429 

4.136 

29  00 

197.572 

98.936 

24.772 

6.293 

10 

298.935 

99,534 

16.571 

4.172 

10 

196.443 

98.924 

24.908 

6.329 

20 

296.357 

99.526 

16.712 

4.208 

20 

195  327 

98.911 

25.044 

6.365 

30 

298.825 

99.518 

16.853 

4.244 

30 

194.223 

98.899 

25.180 

6.400 

40 

291  .334 

99.510 

16.995 

4.280 

40 

193.13', 

98.887 

25.316 

6.436 

50 

288.886 

99.501 

17.136 

4.316 

50 

192.053 

98.874 

25.452 

6.472 

20  00 

286.479 

99.493 

17.277 

4.352 

30  00 

190.986 

98.862 

25.587 

6.508 

310 


TABLE  II.— TANGENT   OFFSETS. 


0 

Curve. 

g 

Curve. 

<J 

1 

.227 

.908: 

51 

.454      .681 

1.135 

2 

.001 

.001 

.001 

.00-2 

5  '2 

.236 

.472      .708 

.944    1.180 

3      .001 

.002 

.00-2 

.003 

.001 

5:! 

.245 

.490,     .735 

.980'   1.225 

4 

.001 

.003 

.004 

.005 

.007 

54 

.254        .509      .763 

1.018    1.272 

5 

.002 

.004 

.007 

.009 

.011 

55 

.264 

.5-28      .792 

1.058.   1.320 

6 

.003 

.006 

.009 

.013 

.010 

56 

.274 

.547      .821 

1.095 

1.368 

7 

.004  i   .009 

.013 

.017 

.021 

57 

.284 

.567      .851 

1.134    1.417 

8 

.006      .011 

.017 

.02-2 

.028 

58 

.294 

.587      .881 

1.174    1.468 

9 

.007      .014 

.021 

.028 

.035 

59        .304 

.608      .911' 

1.215    1.519 

10 

.009      .017 

.026 

.035 

.044!      60         .?14 

.628      .942 

1.250 

1.570 

11 

.011      .021 

.032 

.042 

.053       61      !    .325 

.649      .974 

1.299 

1.623 

12 

.013      .025 

.038 

.050 

.063       62 

.335 

.671    1.006 

1.342    1.077 

13 

.015      .029 

.041 

.059 

.074 

63 

.346 

.693    1.039 

1  385!  1.731 

14 

.017  |    .034 

.051 

.068 

.086 

64 

.357 

.715    1.072 

1.429:   1.787 

15 

.020      .039 

.059 

.079 

.098 

65 

.309 

.737.   1.106 

1.475,  1.843 

16 

.0-22      .045 

.067 

.089 

.112 

66 

.380 

.760    1.140 

1.520    1.900 

17 

.025      .050 

.076 

.101 

.126 

67 

.392 

.783    1.175 

1.567 

1.958 

18 

.028  i    .057 

.085 

.113 

.141 

68 

.404 

.807,  1.210; 

1.614 

2.017 

19 

.032      .063 

.095 

.126        .158 

69 

.415 

.831     1.246 

1.662 

2.076 

20 

.035  i    .070 

.105 

.140        .175 

70 

.428 

.855    1.283 

1.710 

2.137 

21 

.038  !   .077 

.118 

.lot        .192 

71 

.440 

.880    1.320 

1.759 

2.199 

22 

.042      .084 

,1*7 

.169 

.211 

72 

.452 

.905    1  357 

1.809 

2.261 

23 

.046      .092 

.138 

.185 

.231 

73 

.405 

.930J  1  395 

1.860 

2.324 

24 

050 

.101 

.151 

.201 

25 

74 

.478 

.956    1.433 

1.911 

2.388 

25 

.055 

.109 

.1(54 

.218 

'.273 

75 

.491 

.982    1.472 

1.903 

2.453 

26 

.059 

.118 

.177 

.236 

.295 

76 

.504 

1.0081  1.512 

2.016 

2.51<1 

27 

.064 

.127 

.191 

,254 

.818 

77 

.517 

l.,OMU   1.552 

2.069 

2..P81 

28 

.008 

.137       .'2.15 

.'.274 

.342 

78 

.531 

1.062    1.593 

2.123 

2.654 

29 

.073 

.147 

.•220 

.21)4 

.367 

i     7!) 

.515 

1.089    1.634 

2.178 

2.7-2; 

30 

.079 

.157 

.236 

.314 

.393 

;   80 

.559      1.117 

1.675 

2.233 

2.791 

31 

.084 

.168 

.252 

.335 

.419 

81 

.573 

1.145 

1.717 

2.290 

2.86:. 

32 

.089 

.179 

.268 

.357 

.447 

82 

.587 

1.174 

1.700 

2.347 

2.93J- 

33 

.095 

.190 

.285 

.380 

.475 

83 

.601 

1.202 

1.803 

2.404 

3.00: 

34 

.102 

.202 

.303 

.404         .50-1 

81 

.(116 

1.2:11 

1.847 

2.462 

3  07" 

35 

.107 

.214 

.321 

.428        .534 

i     85 

.630 

1.201 

1.891 

2.521 

3.15 

36 

.113 

.226 

.339 

.452 

.565 

86 

.645 

1.991 

1.936 

2  581 

8.22C 

37 

.119 

.239 

.358 

.478 

.597 

87 

.661 

1.321 

1.981 

2.641 

3.30 

38 

.126 

.252 

.378 

.504 

.630 

88 

.676 

1.352 

2.027 

2.702 

3.377 

39 

.133 

.265 

.398 

.531 

.664 

89 

.691 

1.382 

2.  07  3 

2.764 

3.454 

40 

.140 

.279 

.419 

.558 

.69F 

90 

.707 

1.414 

2.  120 

2.8-27 

3.5:5; 

41 

.147 

.293 

.4-10 

.587 

.78.1 

91 

.70: 

1.445 

2.167 

2/90 

3.611 

42 

.154 

.308 

.402 

.616 

.769 

92 

!739 

1.477 

2.215 

2.954 

3.09 

43 

.161 

.323 

.484 

.645 

.so; 

93 

.755 

1.509 

2.263 

8.018 

3.77; 

44 

.169 

.338 

.507 

.675 

.84f 

i     94 

.771 

1.542 

2.313 

3.083 

3.85:- 

45 

.177 

.353 

.530 

.707 

.883 

95 

.788 

1.575 

2.362 

3.149 

8.931 

46 

.185 

.369 

.554 

.739 

.923 

96 

.804 

1.608 

2.412 

3.216 

4.0IJ 

47 

.193 

.386 

.578 

.771 

.964 

97 

.821 

1  612 

2.  -103 

8.283 

4.10; 

48 

.201 

.402 

.603 

.804 

1  .005 

98 

.b3 

1.676 

2.514 

3.351 

4.18!= 

49 

.210 

.419 

.029 

.838 

1.047 

99 

.855 

1.710 

2.565 

3.420 

4  274 

50 

.218 

.436 

.654 

.873 

1.090 

100 

.  873 

1.745 

2.617 

3.489 

4.361 

TABLE  II. -TANGENT  OFFSETS. 


311 


Arc. 

Curve. 

6° 

7° 

8O 

9° 

10° 

11° 

12° 

13° 

14° 

15° 

1 

.001 

.001 

.00! 

.001 

.001 

.001 

.001 

.001 

.001 

.001 

«> 

.002 

.002 

.002 

.003 

.003 

.004 

.004 

.005 

.J005 

.005 

3 

.005 

.005 

.006 

.007 

.008 

.009 

.008 

.010 

.011 

.011 

4 

.008 

.009 

.011 

.013 

.014 

.015 

.017 

.018 

.019 

.020 

5 

.013 

.015 

.017 

.020 

.022 

.024 

.020 

.028 

.031 

.032 

6 

.019 

.022 

.025 

.028 

.03! 

.035 

.038 

.041 

.044 

.047 

,026 

.030 

.034 

.038 

.043 

.047 

.051 

.056 

.oeo 

.064 

8 

.0:54 

.03:) 

.045 

.050 

.056 

.061 

.067!     .073 

.078 

.084 

9 

.042 

.049 

.057 

.064 

'.071 

.078 

.085 

.092 

.099 

.106 

10 

.05,! 

.001 

.070 

.079 

.087 

.090 

.105 

.113 

.122 

.131 

11 

.063 

.074 

.084 

.095 

.106 

.116 

.127 

.137 

.148 

.158 

12 

.075 

.088 

.101 

.113 

.120 

.138 

.151 

.163 

.176 

.188 

13 

.088 

.103 

.118 

.133 

.147 

.162 

.177 

.192 

.206 

.221 

14 

.103 

.120 

.137 

.154 

.171 

.188 

.205 

.222 

.239 

.257 

15 

.118 

.137 

.157 

.177 

.196 

.216 

.236 

.255 

.275 

.294 

16 

.134 

.156 

.179 

.901 

.223 

.246 

.268 

.290 

.313 

.335 

1? 

.151 

.177 

.202 

.227 

.252 

.277 

.303 

.328 

.353 

.378 

18 

.170 

.198 

.220 

.254 

.283 

.311 

.339 

.368 

.396 

.424 

19 

.189 

.221 

.252 

.284 

.315 

.346 

.378 

.409 

.441 

.472 

20 

.209 

.244 

.279 

.314 

.349 

.384 

.419 

.454 

.    .489 

.523 

21 

.231 

.269 

.308 

.346 

.385 

.423 

.462 

.500 

.539 

.577 

22 

.253 

.296 

.338 

.380 

.422 

.465 

.507 

.549 

.591 

.633 

n 

.277 

.323 

.369 

.415 

.462 

.508       .554 

.600 

.646 

.692 

24 

.302 

.352 

.402 

.452 

.503 

.553 

.603 

.653 

.704 

.754 

25 

.327 

.382 

.436 

.491 

.545 

.600 

.654      .709 

.763 

.818 

2G 

.354 

.413 

.472 

.531 

.590 

.649:     .708 

.767 

.826 

.885 

27 

.382 

.445 

..r09 

.572 

.  636 

.7001     .763 

807 

.890 

.954 

28 

.410 

.479 

.547 

.016 

.684 

.752 

.821 

!889 

.957 

.026 

29 

.440 

.514 

.587 

.660 

.734 

.P07 

.880 

.954 

1.027 

.100 

30 

.471 

.550 

.628 

.707 

.785 

.864 

.942 

1.021 

1.099 

.177 

31 

.503 

.587 

.671 

.755 

.838 

.922 

.006 

1.090 

1.174 

.257 

32 

.530 

.625 

.715 

.804 

.893 

.983 

.072 

1.101 

1  250 

340 

.33 

.570 

.665 

.700}     .855 

.950 

.045 

.140 

1.235 

1.330 

.424 

34 

.005 

.706 

.807       .908 

.009 

.1C9 

.210 

1.311 

1.412 

.512 

35 

.641 

.748 

.855       .962 

.069 

.174 

.282 

1.389 

1.496 

.002 

36 

.679 

.792 

.904 

1.018 

.131 

.244 

.357 

1.469 

1.582 

.695 

37 

.717 

.836 

.955 

1.075 

.194 

.314 

.433 

1.552 

1.671 

.790 

38 

.756 

.882 

1.008 

1.134 

.260 

.386 

1.511 

1.637 

1.763 

.P89 

39 

.796 

.929 

1.061 

1.194 

.327 

.459 

1.592 

1.724 

1.857 

.989 

40 

.838 

.977 

1.117 

1.256 

.396 

.535 

1.675 

1.814 

1.953 

2.092 

41 

.880 

.027 

1.173 

1.320 

.466 

.613 

1.759 

1.906 

2.052 

2.198 

42 

.923 

.077 

1.231 

1.385 

.539 

.692 

1.846 

2.000 

2.153 

2.307 

43 

.968 

.129 

1.290 

1.452 

.613!     .774 

1.935 

2.096 

2.257 

2.417 

44 

.014 

.182 

1.351 

1.520 

1.689 

.857 

2.026 

2.194 

2.363 

2.531 

45 

.oeo 

.237 

1.413 

1.590 

1.766 

.943 

2.119 

2.295 

2.472 

2.647 

46 

.108 

.292 

1.477 

1.661 

1.8  1C.    2.030 

2.214 

2.398 

2.582 

2.766 

47 

.15(5 

.349 

1.541 

1.734 

1.9--7    2.119 

2.311 

2.504 

2.696 

2.888 

48 

.206 

.407 

1.608 

1.809 

2.0(10    2.210 

2.411 

2.611 

2.812 

3.012 

49 

.257 

.466J  1.675 

1.885 

2.094    2.303 

2.512    2.721 

2.930 

3.138 

50 

.309 

.527    1  .745 

1.962 

2.180 

2.398 

2.616J  2.833 

3.051 

3.268 

312 


TABLE  II.— TANGENT  OFFSETS. 


Curve. 

Arc. 

16° 

17° 

18° 

19° 

20° 

21° 

2-2° 

23° 

24° 

25° 

1 

.001 

.001 

.002 

.002 

.002 

.002 

.002 

.002 

.002 

.002 

2 

.006 

.006 

.006 

.OC7 

.007 

.007 

.008 

.008 

.008 

.009 

3 

.013 

.013 

.014 

.015 

.016 

.016 

.017 

.018 

.019 

.020 

4 

.022 

.024 

.025 

.027 

.028 

.029 

.031 

.032 

.034 

.035 

5 

.035 

.037 

.039 

.041 

.044 

.046 

.048 

.050 

.052 

.055 

6 

.050 

.053 

.057 

.060 

.063 

.066 

.069 

.072 

.075 

.078 

7 

.068 

.073 

.077 

.081 

.086 

.090 

.094 

.098 

.103 

.107 

8 

.089 

.093 

.101 

.106 

.112 

.117 

.123 

.128 

.134 

.140 

9 

.113 

.120 

.127 

.134 

.141 

.148 

.155 

.163 

.170 

.177 

10 

.140 

.148 

.157 

.165 

.175 

.183 

.192 

.201 

.209 

.218 

11 

.169 

.179 

.190 

.201 

J211 

.222 

.232 

.243 

.253 

.264 

12 

.201 

.211 

.2-26 

.239 

.251 

.264|     .276 

.289 

.302 

.314 

13 

.236 

.251 

.265 

.280 

.294 

.310 

.324 

.339 

.354 

.369 

14 

.274 

.891 

.308 

.325 

.342 

.360 

.376 

.393 

.410 

.427 

15 

.314 

.334 

.353 

.373 

.393 

.412 

.432 

.451 

.471 

.491 

16 

.357 

.379 

.402 

.421 

.447 

.469 

.491 

.514 

.536 

.558 

17 

.403 

.429 

.454 

.179 

.504 

.530 

.555 

.580 

.605 

.630 

18 

.452 

.480 

.509 

.537 

.565 

.594 

.622 

.650 

.678 

.706 

19 

.501 

.535 

.567 

598 

.630 

.661 

.693 

.724 

.756 

.787 

20 

.558 

.593 

.628 

.663 

.698 

.733 

.768 

.802 

.837 

.872 

21 

.616 

.654 

.692 

.731 

.769 

.808 

.846 

.885 

.9-,'3 

.961 

22 

.676 

.718 

.760 

.802 

.844 

.886 

.929 

.971 

1.013 

1.055 

23 

.738 

.784 

.831 

.877 

.923 

.969 

1.015 

1.061 

1.107 

1.153 

24 

.804 

.854 

.904 

.955 

1.005 

1.055 

1.105 

1.155 

1.205 

1.255 

25 

.872 

.927 

.981 

1.036 

1  .090 

1.145 

1.199 

1.253 

1.307 

1.362 

TABLE  III.— TANGENT  OFFSETS  ARC   100  FEET. 


Degree 
of  Curve. 

0' 

10' 

20' 

30' 

40' 

60' 

0° 

.000 

.145 

.291 

.436 

.582 

.727 

1 

.873 

1.018 

1.164 

1.309 

1.454 

1.600 

2 

1.745 

1.891 

2.036 

2.181 

2.327 

2.472 

3 

2.617 

2.763 

2.908 

3.053 

3.199 

3.344 

4 

3.489 

3.635 

3.780 

3.925 

4.070 

4.215 

5 

4.361 

4.506 

4.651 

4.796 

4.941 

5.086 

6 

5.231 

5.376 

5.521 

5.666 

5.811 

5.956 

7* 

6.101 

6.S46 

6.391 

6.536 

6.680 

0.825 

8 

6.970 

7.115 

7.259 

7.401 

7.549 

7.693 

9 

7.838 

7.982 

8.127 

8.271 

8.416 

8.560 

10 

8.705 

8.849 

8.993 

9.137 

9.282 

9.426 

11 

9.570 

9.714 

9.858 

10.01 

10.15 

10.29 

12 

10.43 

10.58 

10.72 

10.86 

11.01 

11.15 

13 

11.30 

11.44 

11.58 

11.73 

11.87 

12.01 

14 

12.16 

12.30 

12.44 

12.59 

12.73 

12.87 

15 

13.01 

13.16 

13.30 

13.44 

13.5!) 

13.73 

16 

13.87 

14.01 

14.16 

14.30 

14.44 

14.58 

17 

14.73 

14.87 

15.01 

15.15 

15.30 

15.44 

18 

15.58 

15.72 

15.86 

16.00 

16.15 

16.29 

19 

16.43 

16.57 

16.71 

16.85 

17.00 

17.14 

20 

17.28 

17.42 

17.56 

17.70 

17.84 

17.98 

21 

18.12 

18.26 

18.40 

18.54 

18.68 

18.82 

22 

18.96 

19.10 

19.24 

19.38 

19.52 

19.66 

23 

19.80 

19.94 

20.08 

20.22 

20.36 

20.50 

24 

20.64 

20.78 

20.02 

21.06 

21.20 

21.33 

25 

21.47 

21.61 

81.75 

21.89 

22.03 

22.16 

26 

22.30 

22.44 

2i!fc8 

22.72 

22.85 

22.  '.(9 

27 

23.13 

23.27 

23.40 

23.51 

28.68 

23.82 

28 

23.95 

24.09 

24.23 

24.30 

24.50 

24.04 

29 

24.77 

24.91 

25.04 

25  J8 

25.32 

25.45 

TABLE  IIIA.-MIDDLE  ORDINATES  ARC  100  FEET. 


313 


Degree 
of  Curve. 

0' 

10' 

20' 

30' 

40' 

50' 

0° 

.000 

.036 

.073 

.109 

.145 

.182 

1 

.218 

.£55 

.291 

.327 

.364 

.400 

2 

.436 

.473 

.509 

.545 

.582 

.618 

3 

.654 

.691 

.764 

.800 

.836 

4 

.873 

.909 

!945 

.9S2 

1.018 

1.054 

5 

.091 

1.127 

1.163 

1.200 

1.236 

1.272 

6 

.309 

1.345 

1.381 

1.418 

1.454 

1.490 

.527 

1.563 

1.599 

1.636 

1.672 

1.708 

8 

.745 

1.781 

1.817 

1.854 

1.890 

l.!26 

9 

.963 

1.999 

2.035 

2.071 

2.108 

2.144 

10 

2.180 

2.217 

2.253 

2.289 

2.325 

2  362 

11 

2.398 

2.434 

2.471 

2.507 

2.543 

2.579 

12 

2.616 

2.  052 

2.688 

2.724 

2.761 

2.797 

13 

2.833 

2.869 

2.906 

2.942 

2.978 

3.014 

14 

3.051 

3.087 

3.1?3 

3.159 

3.195 

3.232 

15 

3.268 

3.304 

8.840 

3.376 

3.413 

3.449 

16 

3.485 

3.521 

3  557 

3.594 

3.630 

3.G67 

17 

3.702 

3.738 

3.774 

3.810 

3.847 

3.8S:5 

18 

3.919 

3.955 

3.991 

4.027 

4.063 

4.100 

19 

4.136 

4.172 

4.208 

4.244 

4.280 

4.316 

20 

4.352 

4.388 

4.42J 

4.460 

4.497 

4.533 

21 

4.  509 

4.605 

4.641 

4.677 

4.713 

4.749 

22 

4.785 

4.821 

4.857 

4.893 

4.929 

4.965 

23 

5.001 

5.037 

5.073 

5.109 

5.145 

5.181 

24 

5.217 

5.253 

5.289 

5.325 

5.361 

5.897 

25 

5.433 

5.468 

5.504 

5.540 

5.576 

5.612 

26 

5.648 

5.684 

5.720 

5.756 

5.792 

5.827 

27 

5.863 

'      5.899 

5.935 

5.971 

8.007 

6.042 

28 

6.078 

6.114 

6.150 

6.186 

6.222 

6.257 

29 

6.293 

0.329 

6.305 

6.400 

6.486 

6.472 

TABLE  IIlB.-CHORDS  FOR  ARCS   100  FEET. 


Degree 
of  Curve. 

0' 

10' 

20' 

80' 

40 

50' 

0° 

100.000 

100.000 

100.000 

iro.ooo 

99.999 

99.999 

1 

99.999 

99.998 

99.998 

99.997 

99.996 

99.996 

2 

99.995 

99.994 

99.993 

99.992 

99.991 

99.990 

3 

99.989 

T.9.987 

99.9H6 

99  984 

09.983 

99.981 

4 

99.980 

99.978 

99.976 

99.974 

99.972 

99  970 

5 

99.968 

99.966 

99.964 

99.962 

99.959 

99.957 

6 

99.954 

99.952 

99.949 

99.946 

99  .  944 

99.941 

99.938 

99.935 

99.932 

99,9-29 

<I9.!>25 

99.922 

8 

99.919 

99.915 

99.912 

99.908 

519  .  HO.") 

99.901 

9 

99.897 

99.893 

99.889 

99.885 

99.882 

90.877 

10 

99.873 

99.869 

99.864 

99.860 

99.856 

99.851 

11 

99.847 

99.842 

99.836 

99.832 

99.827 

99.822 

12 

99.817 

99.812 

99.807 

99.802 

99.797 

99.791 

13 

99.786 

99.780 

99.775 

99.769 

99.763 

99.757 

14 

99.751 

99.745 

99.739 

99.733 

99.727 

99.721 

15 

99.715 

99.708 

99.702 

99.695 

99.689 

99.682 

16 

99.675 

99.669 

99.66.' 

99.655 

99.648 

99.641 

17 

99.634 

99.626 

99.619 

99.612 

99.604 

99.597 

18 

99.589 

99.582 

99.574 

99  566 

99.558 

99.550 

19 

99.542 

99.534 

99.526 

99.518 

9!).  510 

99.501 

20 

99.493 

99.485 

99.476 

91).  467  >  99.459 

99.450 

21 

99.441 

99.432 

99.4-23 

99.414 

99.405 

99.396 

22 

99.387 

99.378 

99.368 

99.359 

99.349 

99  340 

23 

99.330 

99.320 

99.310 

93.301 

99.291 

99.281 

24 

99.271 

99.260 

99.250 

99.240 

99.230 

99.219 

25 

99.208 

99.198 

99.187 

99.177 

99.166 

99.155 

26 

99.144 

99.133 

99.122 

99.111 

99.100 

99.088 

27 

99.077 

99.066 

99.054 

99.043 

99.031 

99.020 

28 

99.008 

98.996 

98.984 

98.972 

98.960 

98.948 

29 

98.936 

98.924 

98.911 

98.898 

98.887 

98.874 

314 


TABLE  IV.— LONG  CHORDS. 


Decree 

2 

3   ! 

4 

5 

6 

n 

8 

9 

10 

ofCurve. 

Sta. 

Sta. 

Sta. 

Sta. 

Sta. 

Sta. 

Sta. 

Sta. 

Sta. 

0°  10' 

200.00 

300.00 

400.00 

499.99 

599.99 

C9J.99 

799.  Of 

699.97 

999.90 

20 

200.00 

300.00 

899.  9'J 

499.98 

599.97 

699.95 

799.93 

899.90 

99'.).  MS 

30 

200.00 

299.99, 

399.98 

499.96 

599.93 

699.89 

799.84 

899.77 

999.68 

40 

200.00 

299.  98  ! 

399.96 

499.93 

599.88 

699.81 

799.71 

8119.59 

99!).  44 

50 

199.99 

299.  98  | 

399.94 

499.89 

599.81 

699.70 

799.55 

899.36 

999.1'.' 

1  00 

199.99 

299.971 

399.92 

409.84 

599.73 

699.56 

799.35 

899.08 

99S.73 

10 

199.99 

299.1:5 

399.89 

499.78 

599.63 

699.41 

799.12 

898.74 

908.27 

20 

199.98 

299.94 

399.86 

499.72 

599.51 

699.23 

798.85 

898.36 

997.75 

30 

199.98 

299.92 

399.82 

499.64 

599.38 

699.02 

798.54 

897.93 

997.15 

40 

199.97 

299  90 

399.77 

499.56 

599.24 

6P8  79 

798  20 

897.43 

996.48 

50 

199.97 

299.88 

399.73 

499.47 

599.08 

698.54 

797.82 

896.89 

995.74 

2  00 

199.96 

299.  86  ! 

399.68 

499.37 

598.90 

698.26 

797.40 

896.30 

994.93 

10 

199.95 

29!).  84 

399.62 

499.26 

598.71 

697.96 

796.% 

895.66 

994.05 

20 

199.94 

299.81; 

399.56 

499.14 

598.51 

697.  K3 

796.47 

894.97 

993.10 

30 

199.94 

299.79 

399.  '49 

499.01 

598.29 

097.28 

795.94 

894.23 

992.09 

40 

199.93 

299.76 

399.42 

498.87 

598.05 

696.91 

795.39 

893.43 

991.00 

50 

199.92 

299.  72  ! 

399.35 

498.73 

597.80 

696.51 

794.78 

892.59 

989.84 

3  00 

199.91 

299.69 

399.27 

498.57 

597.54 

696.09 

794.16 

891.70 

988.62 

10 

199.90 

299.66 

399.19 

498.41 

597.25 

695.64 

793.50 

890.75 

987.32 

20 

199.89 

299.62 

399.10 

498.24 

596.96 

695.17 

792.80 

889.75 

985.96 

30 

199.88 

299.58 

399.01 

498.06 

596.65 

691.68 

79-.'.  06 

888.71 

984.52 

40 

199.86 

299.54 

398.91 

497.87 

596.32 

694.16 

791.29 

887.61 

983.02 

50 

199.85 

299.50 

398.81 

497.67 

595.98 

693.62 

790.48 

886.47 

981.45 

4  00 

199.84 

299.45 

398.70 

497.47 

505.62 

603.06 

789.64 

885.27 

979.82 

10 

199.8-,' 

299.41 

398.59 

497.SJ5 

595.25 

692.47 

788.77 

884.02 

978.1  1 

20 

199.81 

299.36 

398.48 

497.03 

594.87 

691  .85 

787.85 

882.73 

976.34 

30 

199.79 

299.31 

308.36 

496  .  79 

594.40 

691.22 

786.91 

881  .38 

974.50 

40 

199.78 

299.25 

398.23 

496.55 

594.05 

690.56 

785.92 

879.98 

972.59 

50 

199.76 

299.20 

398.11 

496.30 

593.62 

689.87 

784.90 

878.54 

970.61 

5  00 

199.75 

299.14 

397.97 

496.04 

593.17 

689.17 

783.85 

877.05 

968.57 

10 

199.73 

299.09 

397.84 

495.78 

59:.'.  71 

688.44 

782.77 

875.50 

906.46 

20 

199.71 

299.03 

397.69 

495.50 

592.23 

687.68 

781.64 

873.91 

964.29 

30 

199.69 

298.96 

397.55 

495.21 

591.74 

686.90 

780.49 

872.27 

962.05 

40 

199.67 

298.M 

397.40 

494.92 

591.24 

686.10 

7  9.30 

870.58 

959.74 

50 

199.65 

298.84 

397.24 

494.62 

590.72 

685.28 

7  6.07 

868.84 

957.37 

'.  00 

199.63 

298  77 

397.08 

494.31 

590.18 

684.43 

7  6.8! 

867.06 

954.93 

10 

199.61 

298  .'70 

396.92 

493.99 

589.63 

683.56 

7  5.52 

865.22 

952.43 

20 

199.59 

298.63 

396.75 

493.6(5 

589.06 

682.67 

7  4.19 

863.34 

949.86 

30 

199.57 

298.55 

396.58 

493.  8  J 

588.48 

681.75 

7  2.88 

801.41 

947.25 

40 

199.55 

298.48 

396.40 

492.98 

587.89 

680.81 

7  1.43 

859.44 

944.54 

50 

199.53 

298.40 

396.22 

492.62 

587.28 

679.85 

7  0.00 

857.41 

941.78 

7  00 

199.50 

298.32 

396.03 

492.26 

586.66 

678.86 

768.54 

855.34 

938.96 

10 

199.48 

298.24 

395.84 

491.  89 

5S6  .  02 

677.85 

767.04 

853.22 

936.07 

20 

199.45 

£98.16 

395.65 

491.51 

585.36 

676.82 

765.51 

851.06 

93.1.13 

30 

199.43 

298.08 

395.45 

491  .  12 

584.70 

675.77 

763.94 

848.85 

930.12 

40 

199.40 

297.99 

395.24 

490.73 

584.02 

674.69 

762.35 

846.59 

927.05 

50 

199.38 

297.9:) 

395.03 

490.32 

583.32 

673.59 

760.72 

844.29 

923.92 

8  00 

199.35 

297.81 

394.82 

489.91 

582.61 

672.47 

759.05 

841.94 

920.73 

10 

199.32 

297.72 

394.60 

489.49 

581.88 

671.32 

757.36 

839.55 

917.47 

20 

199.30 

:-97.»>3 

394.38 

489.05 

581.14 

670.16 

755.  (13 

837.11 

914.16 

30 

199.27 

297.53 

394.16 

488.62 

580.39 

668.97 

753  87 

834.62 

910.79 

40 

199.24 

297.43 

393.93 

4S8.17 

579.62 

6U7.76 

752.07 

832.09 

907.36 

50 

199.21 

297.33 

393.69 

487.71 

578.84 

666.52 

750.23 

829.52 

903.87 

9  00 

199.18 

297.23 

393.45 

487.25 

578.04 

665.27 

748.39 

826.90 

900.32 

10 

199.15 

297.13 

393.21 

486.77 

577.23 

003.  99 

746.50 

824.24 

896.71 

20 

-.99.12 

297.02 

392.96 

486.29 

576.40 

66.'.  69 

744.58 

821.54 

893.04 

30 

199.08 

296.92 

39->.7I 

485.80 

575.56 

661.37 

742.  0:j 

818.79 

889.32 

40 

199.05 

296.81 

392.45 

485.31 

574.71 

660.02 

740.64 

810.00 

885.55 

50 

199.02 

296.70 

392.19 

484.80 

573.84 

6f,8.6o 

788.68 

813.16 

881.71 

10  00 

198.99 

296.58 

391.93 

484.28 

572.96 

657.27 

736.58 

810.28 

877.82 

TABLE  IV.— LONG  CHORDS. 


|  Degree    2 
or'  •(.  urve  Sta. 

3 

Sta. 

4 
Sia. 

Sta. 

6 
Sta. 

St'a. 

8 
Sta. 

9 
Sta. 

10 
Sta. 

10°  10'  19R.95 

296.47  391.66 

4S3.761  572.06 

655.83 

734.50 

807.37 

873.88 

20 

198.92 

296.  35  j  391.38 

483.23  57J.15 

654.43 

732.39 

804.40 

869.88 

30 

198.88 

296.24:  391.10 

482.69  570.23 

652.98 

730.25 

801  .40 

865.82 

40 

198.85 

296.12;  390.82 

482.14 

569.29 

651.51  728.08 

798.36 

861.72 

50 

11  00 

10 
20 

198.81 

198.77 
198.74 
198.70 

295.99 

295.87 
295.75 
295.62 

390.53 

390.24 
389.95 
389.65 

481  .59 

481.02 
480.45 
479.87 

568.34 

567.37 
566.39 
565.40 

650.01  725.88 

648.50  723.65 
646.96  721.39 
645.4l|  719.10 

795.27 

792.15 

788.98 

785.77 

857.  5C 

853.34 
849.08 
844.76 

30 

198.66 

295.49  389.34 

479.28|  564.39 

643.83!  716.78 

782.53 

840.40 

40 

198.62 

295.36 

389.04 

478.68  563.37J  642.23 

714.44 

779.24 

835.98 

50 

198.58 

295.22 

388.T  2 

478.08  562.  34|  640.61 

712.06 

775.92 

831.51 

12  00 

198.54 

295.09 

388.40 

477.4*  561.29 

638.92 

709.65 

772.55 

826.99 

10 

198.50 

294.95 

388.08 

476.84  560.23  637.31 

707.22 

769.15 

822.43 

20   198.46 

294.81 

387.76 

476.  21  1  559.  16  i  635.63 

704  75 

705.7 

817.81 

30   198.  42 

294.67 

387.43 

475.58  558.07 

633.  9o 

702.26 

762.24 

813.15 

40   198.37 

294.53 

387.09 

474.93;  556.97 

632.21 

699.74 

758.72 

808.44 

50 

198.33 

294.39 

386.76 

474.28  555.86 

630.47 

697.19 

755.17 

803.69 

13  00 

198.29 

294.24 

386.41 

473.621  554.73 

628.7 

694.61 

751.58 

798.89 

10 

198.24 

294.09 

386.07 

472.951  553.59 

626.93 

692.01 

747.95 

794.04 

20 

198.20 

293.94 

385.71 

472.27  552.44 

625.13 

689.37 

744.29 

789.15 

30 

198.15 

293.79 

385.36 

471.58  551.27 

623  31 

686.71 

740.60 

784.21 

40 

198.11 

293.64 

3^5.00 

470.89  550.09 

621.47 

684.03 

736.87 

779  2? 

50 

198.06 

^93.49 

3S4.64 

470.19)  548.90 

619.62 

681.32 

733.10 

774^21 

14  00 

198.02 

293.33 

384.27 

469.48  547.69 

617.74 

678.58 

729.30 

769.15 

10 

1  97.97 

293.17 

833.90 

468.76!  546.47 

615.84 

675.81 

725.46 

764.04 

20 

197.92 

29:5.01 

383.52 

468.04!  545.24 

613.93 

673.02 

721.60 

758.90 

30 

197.87 

292.85 

383.14 

467.30!  544.00 

611.99 

670.20 

717.69 

753.71 

40 

197.82 

292.68 

3S2.75 

466.56  542.74 

610.04 

667.36 

713.76 

748.48 

50   197.77 

292.52 

382.36 

465.82  541.47 

608.07 

664.49 

709.79 

743.22 

15  00  197.72 

292.35  381.97 

465.06  540.19 

606.08 

661.59 

705.79 

737.91 

10 

197.67 

292.18  381.57 

464.30  538.90 

604.07 

658.68 

701.76 

732.57 

20 

197.62 

292.  Oil  381.17 

463.52!  537.59 

602.04:  655.73 

697.70  727.19 

SO 

197.57 

291.83  380.77 

462.74  536.27 

600.00  652.76 

693.61  721.78 

40 

197.52 

291.66!  380.36 

461.96  534.94 

597.93  649.77 

689.48 

716.32 

50 

137.40 

291.  48  |  379.94 

461.16 

533.59 

595.85 

646.75 

685.33 

710.84 

16  00 

197.41 

291.30|  379.53 

460.36 

532.24 

593.75 

643.71 

681.14 

705.32 

10 

197.36 

291.121  379.10 

459.55 

530.87 

591.64 

640.65 

676.93 

699.76 

20 

197.30 

290.94 

378.68 

458.74 

529.49 

589.50 

637.56 

672.69 

694.17 

30 

197.25 

290.76 

378.25 

457.91 

528.10 

587.  35  i  634.45 

668.42 

688.55 

40 

197.19 

290.57 

377.81 

457.08 

526.69 

585.18 

631.32 

664.12 

682.91 

50 

197.14 

290.38 

377.38 

456.24 

625.28 

583.00 

628.16 

659.80 

677.22 

17  00 

197.08 

290.19 

376.93 

455.39 

523.85 

580.80 

624.98 

655.44 

671.50 

10 

197.02 

290.00 

376.49 

454.54 

522.41 

578  .  58 

621.78 

651  .06 

665.76 

20 

196.96 

289.81 

376.04 

453.68 

520.96 

576.34 

618.56 

646.66  659.99 

30 

196.90 

289.61 

375.58 

452.81 

519.49 

573.99 

615.32 

642.23  654.19 

40 

196.85 

289.42 

375.12 

451.93 

518.02 

571.82  612.06 

637.77  648  36 

50 

196.79 

289.22 

374.66 

451.05 

516.53 

569.  53j  608.77 

633.29 

642.50 

18  00 

196.73 

289.02 

374.20 

450.16 

515.04 

567.23 

605.46 

628.78 

636.42 

10 

196.67 

288.82 

373.73 

449.26 

513.53 

564.91 

602.13 

624.25 

20 

196.60 

288.61 

373.25 

448.35 

512.01 

562.58 

598.78 

619.70 

30 

196.54 

288.41 

372.77 

447.44 

510.48 

560.23 

595.42 

615.12 

40 

196.48 

288.20 

372.29 

446.52 

508.93 

557.87 

592.03 

610.52 

50 

196.42 

287.99 

371.80 

445.60 

507.38 

555.49 

588.62 

605.90 

19  00 

196.36 

287.78 

371.33 

444.66 

505.83 

553.09 

585.32 

601.25 

10 

196.29 

287.57 

370.82 

443.72 

504.24 

550.68 

581.75 

596.59 

20 

196.23 

287.35 

370.32 

442.77 

502.65 

548.26 

578.29 

591.90 

30 

196.16 

287.14 

369.82 

441.82 

501.05 

545.81 

574.81 

587.20 

40 

196.10 

286.92 

369.31 

440.86 

499.44 

543.36 

571.31 

582.47 

50 

196.03 

286.70 

368.80 

439.89 

497.83 

540.89 

567.79 

577.72 

20  00 

195.96 

286.48 

368.29 

438.91 

496.20 

538.40 

564.25  592.96 

816 


TABLE  V.— MIDDLE  ORDINATES. 


Deg.  of 

2 

3 

4 

5 

6 

» 

8 

9 

10 

Curve. 

Sta. 

Sta. 

Sta. 

Sta. 

Sta. 

Sta. 

Sta. 

Sta. 

Sta. 

0°  10' 

.145 

.327 

.582 

.909 

1  .309 

1.782 

2.327 

2.945 

3.636 

20 

.291 

.654 

1.164 

1.818 

2.618 

3.5G3 

4.654 

5.889 

7.272 

30 

.436 

.982 

1.745 

2.727 

3.927 

5.345 

6.981 

8.KJ5 

10.91 

40 

.582 

1.309 

2.327 

3.636 

5.235 

7.1-26 

9.307 

11  .7* 

14.54 

50 

1.636 

2.909 

4.545 

6.544 

8.907 

11.68 

11.72 

18.17 

1     00 

.873 

1.963 

3.490 

5.453 

7.852 

10.69 

13.96 

17.C6 

21.80 

10 

.018 

2.291 

4.072 

6.362 

9.160 

12.47 

16.28 

20.60 

•,T>.  43 

','0 

.164 

2.618 

4.653 

7.270 

10.47 

14.25 

18.60 

23.54 

29.06 

30 

.309 

2.945 

5.235 

8.178 

11.77 

16.02 

20.92 

26.48 

32.68 

40 

.454 

3.272 

5.816 

9.086 

13.08 

17.80 

23.24 

29.41 

36.30 

50 

.600 

3.599 

6.397 

9.994 

14.39 

19.58 

25  56 

32.34 

39.91 

2     00 

.745 

3.926 

6.978 

10.90 

15.69 

21.35 

27.88 

35.27 

43.52 

10 

.891 

4.253 

7.560 

11.  SI 

17.00 

23.13 

30.19 

38.20 

47.13 

20 

2.036 

4.580 

8.140 

12.72 

18  30 

24.90 

32.51 

41.12 

50.73 

30 

2.181 

4.907 

8.721 

13.62 

19.61 

26.67 

34  82 

44.04 

54.33 

40 

2.329 

5.234 

9.302 

14.53 

20.91 

28.44 

37.13 

46.95 

57.92 

50 

2.472 

5.561 

9.882 

15.43 

22.21 

30.21 

39.43 

49.86 

61.50 

8    00 

2.617 

5.887 

10.46 

16.34 

23.51 

31.98 

41.73 

52  77 

65.08 

10 

2.763 

6.214 

11.04 

17.24 

24.81 

33.75 

44.04 

filhW 

68.65 

20 

2.908 

6.541 

11.62 

18.15 

26.11 

35.51 

46.33 

58.57 

72.21 

30 

3.053 

6.868 

12.20 

19.05 

27.41 

37.27 

48.63 

61.46 

75.77 

40 

3.199 

7.194 

12.78 

19.96 

28.71 

39.06 

50.92 

64.35 

79.31 

50 

3.344 

7.520 

13.36 

20.86 

30.01 

40.79 

53.20 

67.23 

82.85 

4    00 

3.489 

7.847 

13.94 

21.76 

31.30 

42.55 

55.49 

70.11 

80.38 

10 

3.635 

8.173 

14.52 

22.60 

32.60 

44.30 

57.77 

72.5*8 

89.91 

20 

3.780 

8  499 

15.10 

23.56 

33.89 

46.05 

60.04 

75.84 

93.42 

30 

3.925 

8.826 

15.68 

24.46 

35.18 

47.80 

62.32 

78.70 

96.92 

40 

4.070 

9.152 

16.25 

25.36 

36.47 

49.55 

64  .  58 

81  .55 

100.41 

50 

4.215 

9.478 

16.83 

26.26 

37.76 

51.29 

06.85 

84.39 

103.89 

6    00 

4.351 

9.803 

17.41 

27.16 

39.05 

53.04 

69.11 

87.23 

107.36 

10 

4.506 

10.13 

17.99 

28.06 

40.33 

54.78 

71.36 

90.06 

110.82 

20 

4.651 

10.46 

18.56 

28.96 

41.62 

56.51 

73.61 

92.88 

114.27 

30 

4.796 

10.78 

19.14 

29.85 

42.90 

58.21 

75.86 

95.69 

117.71 

40 

4  941 

11.11 

19.72 

30.75 

44.18 

59.97 

78.10 

98.50 

121.13 

50 

5.086 

11.43 

20  29 

31.61 

45.46 

61.70 

80.33 

101.29 

124.54 

6    00 

5.231 

11.76 

20.87 

32.54 

46.74 

63.43 

82.56 

104.08 

127.94 

10 

5.376 

12.08 

21.44 

33.43 

48.01 

65.15 

84.78 

106.86 

131.32 

20 

5.521 

12.41 

22.02 

34.32 

49.29 

66.86 

87.00 

109.63 

134.69 

30 

5.666 

12.73 

22.59 

35.22 

50.56 

68.58 

89.21 

112.39 

138.05 

40 

5.811 

13.06 

23.17 

36.11 

51.83 

70.29 

91.42 

115.14 

141.39 

50 

5.956 

13:38 

23.74 

36.99 

53.10 

71.99 

93.62 

117.88 

144.71 

7     00 

6.101 

13.71 

24.31 

37.88 

54.37 

73.70 

95.81 

120.62 

148.03 

10 

6.246 

14.03 

24.80 

38.77 

55.63 

75.40 

97.99 

123.34 

151.32 

20 

6.391 

14.35 

25.46 

39.66 

56.89 

77.09 

100.18 

12i;.  05 

154.60 

30 

6.536 

14.68 

26  03 

40.54 

58.15 

78.78 

102.35 

128.75 

157.87 

40 

6.680 

15.00 

26.60 

41.43 

59  41 

80.47 

104.52 

131.44 

161.11 

50 

6.825 

15.33 

27.17 

42.31 

60.67 

82.15 

106.67 

134.11 

164.34 

8    00 

6.970 

15.65 

27.74 

43.19 

61.92 

83.83 

108.83 

136.78 

167.56 

10 

.115 

15.97 

28.31 

44.07 

63.17 

85.51 

110.97 

139.44 

170.75 

20 

.259 

16.30 

28.88 

44.95 

64.42 

87.18 

113.11 

142.08 

173.93 

50 

.404 

16.62 

29.45 

45.83 

65.60 

88;.  84 

115.24 

144.71 

177.09 

40 

.549 

16.94 

30.02 

46.71 

66.91 

90.50 

117.36 

147.33 

180.23 

50 

.693 

17.27 

30.59 

47.58 

68.15 

92.16 

119.48 

149.94 

183.36 

9    00 

.888 

17.59 

31.16 

48.46 

69.39 

93.81 

21.58 

152.53 

186.46 

10 

.982 

17.91 

31.73 

49.33 

70.62 

95.46 

23.68 

155.11 

189.55 

20 

8.127 

18.23 

:52.29 

50.21 

71.86 

97.10 

25  .  77 

157.68 

19'.'.  61 

30 

8.271 

18.56 

32.86 

51.08 

73  09 

98.74 

27'  .  Sf) 

100.23 

195.06 

40 

8.416 

18.88 

33.42 

51.95 

74.81 

100.37 

89  JOB 

162.77 

19S.OH 

50 

8.560 

19.20 

33.99 

52.81 

75.54 

102.00 

131.99 

165.30 

201.08 

10    00 

8.705 

19.52 

34.55 

53.68 

76.76 

103.62 

134.05 

167.82 

204.67 

TABLE  V.— MIDDLE  ORDINATES. 


317 


Deg.  of 
Curve. 

2 

Sta. 

3 

Sta. 

4 

Sta. 

5 
Sta. 

6 
Sta. 

St'a. 

8 
Sta. 

9 
Sta. 

10 
Sta. 

10°  10' 

8.849 

19.84 

35.12 

54.55 

77.98 

105.23 

136.09 

170.31 

207.63 

20 

8.993 

20.17 

35.68 

55.41 

79.20 

106.85 

138.13 

172.80 

210.57 

30 

9.137    20.49 

36.24 

56.27 

80.41 

108.45 

140.16 

175  A'  7 

213.  4<i 

40 

9.282;  20.81 

36.81 

57.14 

81.62 

110.05 

142.18!   177.73 

216.30 

50 

9.426    21.13 

37.37 

57.99 

82.83 

111.64 

144.19 

180.17 

219.26 

11    00 

9.570    21.45 

37.93 

58.85 

84.03 

113.23 

146.19 

182.59 

222.il 

10 

9.714    21.77 

38.49 

59.71 

85.23 

114.82 

148.18 

185.  CO 

224.94 

20 

9.858    22.09 

39.05 

60.56 

86.43 

116.39 

150.16 

187.40 

227.75 

30 

10.00 

22.41 

39.61 

61.42 

87.62 

117.96 

152.13 

189.78 

230.53 

40 

10.15 

22.13 

40.16 

62.27 

88.82 

119.58 

154.09 

192.14 

233.2? 

50 

10.29 

23.05 

40  72 

63.12 

90.00 

121.09 

156.04 

194.49 

230.02 

12    00 

10.43 

23.37 

41.28 

63.97 

91.19 

122.64 

157.98 

196.82 

238.73 

10 

10.58 

23.69 

41.84 

64.81 

92.37 

124.19 

159.91 

20 

10.72 

24.01 

42.39 

65.66 

93.55 

125.73 

161.83 

30 

10.86 

24  33 

42.95 

66.50 

94.72 

127.26 

163.73 

40 

11.01 

24.64 

43.50 

67.35 

95.89 

128.79 

165.63 

50 

11.15 

24.96 

44.05 

68.19 

97.06 

130.31 

167.52 

13    00 

11.30 

25.28 

44.61 

69.02 

98.22 

131.82 

169.39 

10 

11.44 

25.60 

45.16 

69.86 

99.38 

133.33 

171.26 

20 

11.58 

25.92 

45.71 

70.69 

100.54 

134.83 

173.11 

30 

11.73 

26.23 

46.26 

71.53 

101.69 

136.32 

174.95 

40 

11.87 

26.55 

46.81 

72.36 

102.83 

137.81 

176.78 

50 

12.01 

26.87 

47.36 

73.19 

103.98 

139.29 

178.60 

14    00 

12.16 

27.18 

47.90 

74.01 

105.12 

140.76 

180.40 

10 

12.30 

27.50 

48.45 

74.84 

106.26 

142.23 

20 

12.44 

27.81 

49.00 

75.66 

107.39 

143.68 

30 

12.59 

28.13 

49.54 

76.48 

108.52 

145.13 

40 

12.73 

28.45 

50.09 

77.30 

109.64 

146.58 

50 

12.87 

28.76 

50.63 

78.12 

110.76 

148.01 

15    00 

13.02 

29.08 

51.17 

78.93 

111.88 

149.44 

10 

13.16 

29.39 

5].  72 

79.  15 

112.99 

150.86 

20 

13.30 

'~:!).70 

52.26 

80.56 

114.10 

lfia.28 

30 

13.44 

30.0.2 

52.80 

81.37 

115.20 

153.68 

40 

13.59 

30.33 

53.34 

82.17 

116.30 

155  08 

50 

13.73 

30.65 

53.88 

82.98 

117.  o9 

156.47 

16    00 

13.87 

30.96 

54.41 

83.78 

118.48 

157.85 

10 

14  01 

31.27 

54.95 

84.58 

119  57 

20 

14.16 

31.58 

55.49 

85.38 

120.65 

30 

14  30 

31.90 

56.02 

86.17 

121.73 

40 

14.44 

32.21 

56.55 

86.97 

122.80 

50 

14.58 

32.52 

57.09 

87.70 

123.87 

17    00 

14.73 

3-2.83 

57.62 

88.55 

124.93 

10 

14.87 

33.14 

58.15 

89  33 

125.99 

20 

15.01 

33.45 

58.68 

90.12 

127.04 

30 

15.15 

33.70 

59.21 

£0.90 

128.09 

40 

15.30 

34.07 

59.74 

91.68 

129.14 

50 

15.44 

34.38 

00.27 

92.46 

130.18 

18    00 

15.58 

34.69 

fO.79 

93.23 

131.21 

10 

15.72 

35.00 

61.32 

94.00 

132.24 

20 

15.86 

35.31 

61.84 

94.77 

133.27 

30 

16.00 

35.62 

655.86 

95.54 

184.29 

40 

16.15 

35.93 

62.89 

96.31 

135.30 

50 

10.29 

36.24 

03.41 

97.07 

130.31 

19    00 

16.43 

36.54 

63.93 

97.83 

137.32 

10 

16.57 

30.85 

(4.44 

98.59 

138.32 

20 

16.71 

37.16 

04.96 

99.34 

139.31 

30 

10.85 

37.46 

65.48 

ICO.  09 

140.30 

40 

16.99 

37.77 

(.6.00 

100.84 

141.29 

50 

17.14 

38.08 

66.51 

101.59 

142.27 

20    00 

17.28 

38.38 

67.02 

102.33 

143.24 

TABLE  VI. -TURNOUTS  FROM  A  STRAIGHT  TIlACMv. 


Switch-rail  15  Feet,  thrown  5  inches. 

No. 

Angle 

Radius. 

Degree 

Lead 

No. 

Angle 

Diet. 

7    771 

ii. 

^'. 

C'enter. 

of  Curve. 

1-1  F. 

1<". 

b  "  . 

H'F". 

Jijl1  . 

0              / 

~~7      77 

0              / 

4 

14     15 

138.81) 

41   15  09 

45.84 

2.95 

19     16       31.71      34.77 

4^ 

12    41 

170.43 

8-.'  28  80 

-S9.2S 

3.31 

17     11        36.71      39.2-i 

5 

11     25 

218.09 

26  11  59 

52.63 

3.66 

15     33        3S.64      43.  7  i 

t>H 

10    23 

265.75 

21  33  m 

55.90 

4  03 

14    09       40.57 

48.85 

0 

9    32 

317.6*1 

18  21  25 

59.08 

4.38 

13     01        42  37 

62.95 

6H 

8    48 

37-1.75 

15  17  21 

62.20 

4.74 

12    03 

44.11 

67.  6f 

8     10 

-i  37.06 

13  06  81 

65.25 

5.09 

11     14 

45.78 

C2.4C 

7^ 

7    38 

50J.CS 

11  21   08 

6.S-.2 

5.43 

10    31        47.38 

67.  3l 

8 

7    09 

577.85 

9  54  55 

71.12 

5.78 

9    54  !     4S.92 

72.24 

8^j 

6    44 

650.79 

8  43  24 

73.98 

6.12 

9    21 

50.40 

1   *   .  i»C 

9 

0    22 

741.67 

7  43  50 

76.75 

6.45 

8    52 

61.60 

82.42 

9^ 

6    02 

(582.75 

6  52  48 

19.47 

6.78 

8     26 

53.20 

87.  «6 

10 

10/4 

5    43 

930.4'J 
1034  5C 

C  09  29 
5  32  19 

82.12 
84   r'\ 

7.11 
7  4A 

8    03 

54.52 

'.i.'i  <>: 

li 

5     12 

1145^90 

5  00  GO 

O-t  .  I  O 

87.  '.7 

t  .4«> 

7.75 

7     23 

57.01 

104.17 

1H* 

4    59 

1264.40 

4  31  53 

88.76 

8.07 

7    06 

58.  IS  ilt".l.«K 

12 

4    46 

1390.90 

4  07  10 

92.  20 

S.37 

6     50        59.30  !  115.9(i 

Switch-rail  8  Feet,  thrown  5  inches. 

No. 
n. 

Angle 

Radius, 

("enter. 

Degree 
of  Curve. 

Lead 
11  F. 

No. 
n''. 

Angle 

JT  '  '  * 

Dist. 
H'F. 

LF. 

3 

18  55  28 

78.08 

72  27  1  1 

30.17 

2.19     '<  25    45 

21.91 

26.41 

3J4 

16  15  32 

108.73 

52  41   44 

33.31 

2.80        22     15        23.65 

31.10 

4 

14  15  00 

143.56 

39  54  26 

36.32 

2.89     1   19     38  i     25.27 

35.92 

41^ 

12  40  48 

183.98 

31  03  4S 

39.20 

3.23 

17    35 

26.78 

40.90 

5 

11  25  16 

230.31 

24  52  32 

41.96 

3.57 

15    58 

28.18 

46.07 

5}»12 

10  23  20 

283.05 

20  14  16 

-1-1.60        3.89 

14     38 

89.49 

51  .47 

6 

931  39 

342.70 

16  43  ',".) 

47.15       4.21        13    32 

30.70 

57.12 

6J^ 

8  47  51 

409.97 

13  58  28    |     49.59  ,     4.52        12     37 

31.84 

63  07 

TABLE  V1T.-TANCKNTS  AND  EXTERNALS  OF  A  V  CURVE.  319 


Angle 

Tan^. 
T. 

Ex. 

E. 

Angle 
V. 

Tang. 
T. 

Ex. 

E. 

A  ngle 

Tang. 

Ex. 

E. 

90°  10' 

34477(5 

143105 

100°  10' 

410907 

191913 

110°  10' 

492484 

256826 

20 

345781 

142817 

20 

412123 

19^-01 

20 

494013 

258081 

30 

346788 

144531 

30 

413324 

193845 

30 

495549 

259342 

40 

347798 

145249 

40 

414569 

194781 

40 

497091 

260610 

50 

348811 

145971 

50 

415798 

JUB734 

£0 

498767 

261884 

91  00 

349828 

146695 

101  00 

417032 

196685 

111  00 

500195 

263165 

10 

350847 

147422 

10 

418270 

191641 

10 

501757 

264453 

20 

351809 

148153 

20 

419512 

198601 

20 

503326 

265748 

30 

352895 

14S887 

30 

420759 

199566 

30 

504901 

261049 

40 

353!i23 

149624 

40 

422010 

200536 

40 

506483 

268351 

50 

354955 

150365 

50 

423206 

201510 

50 

508071 

269673 

92  00 

355989 

151108 

102  CO 

424526 

202489 

112  00 

508667 

270995 

10 

357027 

151855 

10 

425191 

203472 

10 

511269 

272324 

20 

3580138 

152606 

20 

427060 

','044  CO 

20 

512879 

213660 

30 

359112 

153359 

30 

428334 

205453 

30 

514495 

275C03 

40 

360159 

154116 

40 

429613 

206451 

40 

516119 

276354 

50 

361209 

154877 

50 

430896 

207453 

50 

517749 

217712 

98  00 

362203 

155640 

103  00 

432184 

208461 

113  CO 

519387 

279077 

10 

363320 

156408 

10 

433477 

209473 

10 

521032 

280449 

20 

364380 

157178 

20 

434774 

210490 

20 

522684 

281829 

30 

365443 

157952 

30 

436016 

211512 

30 

524344 

283216 

40 

366510 

158730 

40 

437383 

212539 

40 

526010 

284611 

50 

3675b'0 

159511 

50 

438695 

213571 

50 

527685 

286013 

94  00 

368653 

160295 

104  CO 

44C012 

214608 

114  00 

529367 

287423 

10 

369730 

161083 

10 

441333 

215650 

10 

531056 

288840 

20 

370810 

161874 

20 

442660 

216697 

20 

532753 

290265 

30 

371898 

162670 

30 

443991 

217749 

30 

534457 

291698 

40 

372i'80 

163468 

40 

445328 

218807 

40 

526170 

293139 

50 

3740:  0 

164270 

50 

446069 

219869 

50 

537890 

294588 

95  00 

375164 

165076 

105  00 

448016 

220937 

115  00 

539618 

296045 

10 

316261 

165886 

10 

449368 

222010 

10 

541354 

297509 

20 

SI  736-2 

166699 

20 

450725 

223088 

20 

548098 

299051 

30 

37K4r,6 

167516 

30 

452087 

224172 

30 

544850 

300463 

40 

379574 

168336 

40 

453454 

225261 

40 

546610 

301952 

50 

380686 

169161 

50 

454827 

226355 

50 

548378 

303450 

96  00 

381800 

169989 

'106  00 

456204 

221455 

116  00 

550155 

304956 

10 

382919 

170820 

'10 

457588 

228560 

10 

551939 

306412 

20 

384041 

171656 

20 

458976 

228671 

20 

553732 

307992 

30 

385167 

172495 

30- 

460310 

230788 

30 

555534 

309522 

40 

386297 

173338 

40 

461770 

232909 

40 

557344 

311063 

50 

387430 

174186 

50 

463174 

233037 

50 

559162 

312612 

97  00 

388567 

175036 

107  00 

464585 

234170 

117  00 

560989 

314169 

10 

389707 

175891 

10 

466001 

235309 

10 

562825 

315735 

20 

390852 

176750 

20 

467422 

236453 

20 

564670 

317310 

30 

392000 

177613 

SO 

46C850 

237604 

30 

566523 

318F94 

40 

393152 

178480 

40 

410282 

238760 

40 

568386 

820487 

50 

394308 

179350 

50 

471721 

239922 

50 

570257 

322089 

98  00 

395468 

180225 

108  00 

473165 

241090 

118  00 

572137 

323700 

10 

396631 

181104 

10 

474615 

242264 

10 

574027 

325320 

20 

397799 

181987 

20 

476071 

243443 

20 

515925 

326950 

30 

398970 

182873 

30 

477533 

244629 

30 

511833 

328589 

40 

400146 

183764 

40 

479001 

245821 

40 

579151 

330237 

50 

401325 

184660 

50 

480475 

247019 

50 

581677 

331895 

99  00 

402508 

185559 

109  CO 

481954 

248223 

119  00 

583614 

3335C3 

10 

403696 

186462 

10 

483440 

249433 

10 

585559 

335240 

20 

40-1887 

187370 

20 

484932 

250649 

20 

587515 

336927 

30 

406083 

188282 

30 

486430 

251872 

30 

589480 

338624 

40 

407283 

189198 

40 

481934 

253101 

40 

591455 

340331 

50 

408487 

190119 

50 

489444 

254336 

50 

593440 

342048 

100  00 

409695 

191044 

110  00 

490961 

255578 

120  00 

595435 

343775 

320    TABLE  VII.— TANGENTS  AND  EXTERNALS  OF  A  1'  CURVE. 


Ansle 

Taiig.       Ext. 

'  Angle 

Tang. 

Ext. 

Angle 

Tang. 

Ext. 

//; 

T.      j      E. 

Pi 

T. 

E. 

/'. 

T. 

E. 

60°  10' 

199146 

53516.2 

70°  10' 

241460 

76325.0 

80°  10' 

289314 

105540 

20 

199814 

53851.6 

20 

242207 

76755.0 

20 

290169 

106091 

30 

200483 

54188.4 

30 

242956 

77186.8 

30 

291027 

106645 

40 

201154 

fi45-26.7 

40 

243706 

77620.4 

40 

291886 

107200 

50 

201826 

54866.4 

50 

244458 

78055.8 

50 

292748 

107759 

61  00 

202499 

55207.5 

71   00 

245212 

78493.0 

81   00 

293611 

108319 

10 

203173 

55550.0 

10 

245967 

78932.0 

10 

294477 

108S82 

20 

203818 

55894.0 

20 

246724 

79372.8 

20 

295345 

109447 

30 

204524 

56239.4 

30 

247482 

79815.4 

30 

296215 

110014 

40 

20520  1 

56586.3 

40 

248242 

80259.8 

40 

297088 

1105S4 

50 

205881 

56934.6 

50 

249004 

80706.1 

50 

297962 

111157 

62  00 

206561 

57284.3 

72   00 

249767 

81154.2 

82   00 

298839 

111731 

10 

20724-2 

57635.6 

10 

250532 

81604.1 

10 

299718 

11230S 

20 

207924 

57988.3 

20 

251298 

82056.0 

20 

300599 

11288S 

30 

208(508 

58342.4 

30 

252066 

82509.6 

30 

301482 

113470 

40 

209-292 

58698.1 

40 

252836 

82965.2 

40 

302368 

114054 

50 

209978 

59055.2 

50 

253607 

83422.6 

50 

303-256 

114641 

63   00 

210065 

59413.8 

73  00 

254380 

83877.5 

83  00 

304146 

115931 

10 

211354 

59773.9 

10 

255154 

84343.1 

10 

305039 

115823 

20 

212043 

60135.5 

20 

255931 

84806.2 

20 

305933 

116417 

30 

212734 

60498.6 

30 

256709 

85271.3 

30      306831 

11701J 

40 

213426 

60863.2 

40 

257488 

85738.2 

40 

307730 

117613 

.50 

214120 

61229.3 

50 

258370 

86207.0 

50 

308632 

118215 

64  00 

214814 

61597.0 

74   00 

259053 

86677.9 

84  00 

309536 

118820 

10 

215510 

61966.2 

10 

259838 

87150.6 

10 

310443 

119427 

20 

216207 

62336.9 

20 

260624 

87625.3 

20 

311352 

120087 

30 

216900 

62709.1 

30 

261412 

88102.0 

30 

312263 

120649 

40 

217605 

63082.9 

40 

262202 

88580.6 

40 

313177 

121264 

50 

218306 

63458.3 

50 

262994 

89061.2 

50 

314093 

121881 

65   00 

219009 

63835.2 

75  00 

263788 

89543.8 

85  00 

315011 

122501 

10 

219712 

61213.7 

10 

264583 

90028.4 

10 

315933 

123124 

20 

220417 

64593.7 

20 

265380 

90515.0 

20 

316856 

123749 

30 

22  11  A3 

64975  .3 

30 

266179 

91003.6 

30 

317782 

124378 

40 

221831 

65358.5 

40 

266979 

91494.3 

40 

318711 

125008 

50 

222540 

65743.3 

50 

267782 

91987.0 

50 

3196)2 

125642 

60  00 

223250 

66129.6 

76  00 

208586 

92481.7 

86  00 

320575 

126278 

10 

2)3961 

66517.  0 

10 

269392 

92978.4 

10 

321511 

12fi9i7 

20 

224674 

66907.2 

20 

270200 

93477.2 

20 

322450 

127558 

30 

225389 

67298.3 

30 

271010 

9397'8.1 

30 

323391 

128203 

40 

226104 

67691.2 

40 

271822 

94481.1 

40 

324335 

128850 

50 

226821 

68085.6 

50 

272635 

94986.1 

50 

3-25281 

129500 

67   00 

227540 

68481.6 

77   00 

273451 

95493.3 

87   00 

326230 

130153 

10 

228259 

68879.3 

10 

274268 

96002.5 

10 

327182 

130808 

20 

228980 

69278.6 

20 

275087 

96513.9 

20 

32H136 

131466 

30 

229703 

69679.6 

30 

275908 

97027.4 

30 

:;290!)3 

1321-J8 

40 

230427 

70082.3 

40 

276731 

97543.0 

40 

330052 

132792 

50 

231152 

70486.5 

50 

277556 

98060.8 

50 

831014 

133159 

68  00 

231879 

70892.5 

78  00 

278383 

98580.7 

88  00 

331979 

134128 

10 

232607 

71300.1 

10 

279212 

99102.8 

10 

332947 

134801 

20 

233337 

71709.5 

20 

280043 

99627.0 

20 

333917 

135477 

30 

234068 

721?0.5 

30 

280876 

100153    ! 

30 

334890 

136155 

40 

234800 

72533.2 

40 

281710 

100682 

40 

335866 

136K37 

50 

235531 

729*7.6 

50 

282547 

101213 

50 

336845 

137521 

69   00 

236270 

73363.7 

79  00 

283386 

101746 

89   00 

337826 

138-208 

10 

887007 

73781.5 

10 

284227 

102281 

10 

338811 

13S8H9 

20 

237745 

74201.1 

20 

285070 

102819 

20 

339798 

139592 

30 

238485 

74622.4 

30 

285914 

103358 

30 

34078,8 

140289 

40 

239226 

75045.4 

40 

286761 

103901 

40 

341780 

140988 

50 

239969 

75470.2 

50 

287610 

104445 

50 

342776 

141G91 

70  00 

240714 

75896.7 

80  00 

288461 

104991 

90  00 

343775 

14039') 

TABLE  VII.— TANGENTS  AND  EXTERNALS  OF  A  1'  CURVE.  321 


Angle 
K 

Tang. 
T. 

Ex. 
ff. 

Angle 
V. 

Tang. 
T. 

Ex. 
E. 

Angle 

Tang. 
T. 

Ex. 

E. 

0°  10' 

500.0 

.304 

10°  10' 

30580.3 

1357.4 

20°  10' 

61132.4 

5393.2 

20 

1000.0 

1.454 

20 

31084.3 

1402.5 

20 

61648.4 

5483.9 

30 

1500.0 

3.273 

30 

31588.5 

1448.2 

30 

62164.6 

5575.4 

40 

2000.0 

5.818 

40 

82092.7 

1494.7 

40 

62681.1 

5667.6 

50 

2500.0 

9.090 

50 

32597.2 

1542.0 

50 

63197.8 

5760.7 

1     00 

3000.1 

13.090 

11    00 

33101.7 

1590.0 

21    00 

63714.9 

5854.6 

10 

3500.1 

17.818 

10 

33606.5 

1638.7 

10 

64232.2 

5949  2 

20 

4000.2 

23.272 

20 

34111.3 

1688.2 

20 

64749.8 

6044  7 

80 

4500.3 

29.455 

30 

34616.3 

1738.4 

30 

65267.7 

6140.9 

40 

5000.4 

36.364 

40 

35121.4 

1789.4 

40  I  65785.8 

6237.9 

50 

5500.5 

44.002 

50 

35626.7 

1841.2 

50  I  66304  .  3 

6335.7 

2     00 

6000.6 

52.367 

12    00 

36132.2 

1893.6 

22    00 

668-J8.0 

6434.3 

10 

6500.  S 

61.459 

10 

36637.8 

1946.8 

10 

67342.1 

6533.7 

20 

7001.0 

71.280 

20 

37143.5 

2000.8 

20 

67861.4 

6633.it 

30 

7501.2 

81.829 

30 

37649.4 

2055.5 

30 

68381.1 

673r>.  0 

40 

8001.4 

93.105 

40 

38155.5 

2110.9 

40 

68901.0 

6836.8 

50 

8501.7 

105.11 

50 

38661.8 

2167.2 

50 

69421.2 

6939.  4 

8    00 

9002.1 

117.84 

13   00 

39168.2 

2224.1 

23   00 

699H.8 

7042.8 

10 

9502.4 

131.31 

10 

39674.8 

2281.8 

10 

70462.6 

7M7.0 

20 

10002.8 

145.50 

20 

40181.5 

2340.3 

20 

70983  .  8 

7252.0 

30 

10.103.3 

160.41 

30 

40688.4 

2399  5 

30     71505.2 

7357.8 

40 

11003.8 

176.06 

40 

41195.5 

2459.5 

40 

72027.3 

7464.4 

50 

11504.3 

192.44 

50 

41702.8 

2520.2 

50 

72549.1 

7571.9 

4    00 

12004.9 

209.55 

14   00 

42210.2 

2581.7 

24   00 

73071.6 

7680.1 

10 

12505.5 

227.38 

10 

42717.9 

SS43.9 

10 

73594.3 

7789.2 

20 

13006.2 

245.95 

20 

43225.7 

2706.9 

20 

74117.4 

789U.O 

30 

13506.9 

265.24 

30 

43733.7 

2770.7 

30 

74640.8 

8009.7 

40 

14007.7 

285.27 

40 

44241  8 

2835.1 

40 

75164.6 

8I21.2 

50 

14508.6 

30(5.02 

50 

44750.2 

2900.4 

50 

75688,6 

8233  .  5 

5    00 

15009.5 

327.51 

15   00 

45258.8 

296G.4 

25   00 

76213.0 

8346.7 

10 

15510.5 

349.72 

10 

45767.6 

3033.2  1 

10 

76737.8    8460.6 

so 

16011  6 

372.67 

20 

46276.5 

3101.7 

20 

77262.81  8575.4 

30 

16512.7 

396.35 

30 

46785.7 

3169.0 

30 

77788.3    8691.0 

40 

17013.9 

420.76 

40 

47295.0 

3238  1 

40 

78314.0 

8807.4 

50 

17515.1 

445.90 

50 

47804.6 

3307.9 

50 

78840.2 

8924.6 

6    00 

18016.5 

471.78 

16  00 

48314.4 

3378.5 

26   00 

79366.6 

9042.7 

10 

18517.9 

498.38 

10 

48824.4 

3449.8 

10 

79893.5    9161.6 

20 

19019.4 

525.72 

20 

49334.6 

3521.9 

20 

80420.7    9281.3 

30 

19520.9 

553.79 

30 

49845.0 

*594.8 

30 

80948.2!  9401.8 

40 

20022.6 

582.60 

40 

50355.6 

3608.4  i 

40 

81476.1 

9523.2 

50 

20524.3 

612.13 

50 

50806.  4  ,  3742.8 

50 

82004.4 

9645.4 

7     00 

21026.2 

642.41 

17   00 

51377.  5'  8818.0  ! 

27   00 

82533.0 

9768.4 

10 

21528.1 

673.41 

10 

51888.8!  3893.9  : 

10 

83062.0!  9892.3 

20 

22030.1 

705.15 

20 

52400.3    3970.7 

20 

83591.4:10017.0 

30 

22532.2 

737.62 

30 

52912  0    4048.1 

30 

84121.1  110142.  5 

40 

23034.4 

770.83 

40 

53424.0    4126.4 

40 

84651.  31  10268.9 

50 

23536.7 

804.78 

50 

53936.2:  4205.4 

50 

85181.8 

10396.1 

8    00 

24039.1 

839.46 

18   00 

54448.5    4285.2 

28  00 

85712.7 

10524.2 

10 

24541.6 

874.88 

10 

54961.2    4365.8  i 

10 

86243.9 

10653.1 

20 

25044.2 

911.03 

20 

55474.1:  4447.1  ; 

20 

86775.6 

10782.8 

30 

25546.9 

947.92 

30 

55987.3    4529.2 

30 

87307.6 

10913.4 

-JO 

26049.'; 

985.55 

40 

56500.6    4612.1  i 

40 

87840.1 

11044.9 

50 

26552.6 

1028.91 

50 

57014.3    4695.8  j 

50 

88372.9 

11177.1 

9    00 

27055.7 

1063.91 

19   00 

57528.  2'  4780.2 

29   00 

88906.2 

11310.3 

10 

27558.8 

1102.86 

10 

58042  3    4865.4 

10 

89439.8 

11444.3 

20 

28062.1 

1143.43 

20 

58556.7    4951.5 

20 

89973.9 

11579.1 

30 

2S5I55.5 

1184.75 

30 

59071.3    5038.2 

30 

90508.3 

11714.8 

40 

29069.0 

1226.83 

40 

50,^6.2    5125.8  ' 

40 

91043.2 

11851.4 

50 

2'.)572.fi 

1269.63 

50 

60101.3    5214.1 

'.,0 

91578.5 

11988.8 

10     00      30076.4 

1H1S.15 

20  00 

60616.8'  5303.3 

30  00 

92114.2 

12127.1 

322  TABLE  VII.-TANGENTS  AND  EXTERNALS  OF  A  1'   CURVE. 


Angle 

Tang.       Ext. 
T.           E. 

Angle 

Tang. 

Ext. 
E. 

Angle 

]/  , 

Tang. 

Ext.   ! 
E.      i 

30°  10' 

92650.3;  12->66.2 

40°  10' 

125690 

22256.8 

:  60°  10' 

160914 

35796.5 

20 

93186.8 

12406.2 

20 

126257 

22452.0 

20 

101524 

36055.5 

30 

93723.7 

12547.1 

30 

126825 

22648.1 

30 

102135 

36315.7 

40 

94261.1 

12688.8 

40 

127394 

22845.2 

40 

102746 

36577  0 

50 

94798.9 

12831.4 

50 

127963 

23043.3 

50 

163359 

36839.4 

31   00 

95337.1 

12974.8 

41   00 

128532 

23242.4 

51   00 

1031)72 

37103.1 

10 

95880.5 

13119.2 

10 

129102 

23442.5 

10 

164586 

37307.9 

20 

96414.9 

13264.3 

20 

129673 

23643.6 

20 

105201 

37633.9 

30 

96954.5 

13410.4 

30 

130245 

23845.6 

30 

165817 

37901  .  1 

40 

97494.5 

13557.4 

40 

130817 

24048.7 

40 

100434 

38169.4 

50 

98034.9 

13705.2 

50 

131389 

SM352.8 

50 

167054 

384:59.0 

32   00 

98575.8 

13853.9 

42  00 

131963 

24457.9 

52  00 

167670 

38709.7 

10 

99117.2 

14003.5 

10 

132537 

240(53.9 

10 

168290 

38981.0 

20 

99658.9 

14151.0 

20 

133111 

24871  0 

20 

168910 

39254  7 

30 

100201 

14305.4 

30 

133687 

25070.2 

30 

169531 

39529.1 

40 

100744 

14457.6 

40 

134203 

25288  3 

40 

170158 

3H804.1! 

50 

101287 

14610.7 

50 

134839 

25498.5 

50 

170770 

40081.3 

33  00 

101831 

14764.7 

43  00 

135416 

25709.6 

63  00 

171400 

40359.3 

10 

102375 

14919.7 

10 

135994 

25921  .  S 

10 

172024 

4063S.4 

20 

102919 

15075.5 

20 

136573 

26135.0 

20 

1  72050 

40918  8 

30 

103464 

15232.2 

30 

137152 

26349.3 

30 

173277 

41200.4 

40 

104010 

15383.  8 

40 

137732 

215564  6 

40 

173904 

41483.3 

50 

104556 

15548.3 

50 

138313 

26780.9 

50 

174533 

41707.4 

34  00 

105102 

15707.7 

44  CO 

138894 

26998.2 

64   00 

175162 

42052.7 

10 

105649 

15868.0 

10 

139476 

27216.0 

10 

175792 

42339.2 

20 

106197 

16029.2 

2u 

140059 

27436.1 

20 

176423 

42627.0 

30 

106745 

16191.3 

30 

140642 

27656.6 

30 

177056 

429  Hi.  (1 

40 

107293 

10354.3 

40 

141226 

27878.1 

40 

177689 

43200.3 

50 

107842 

16518.2 

50 

141811 

28100.7 

50 

178323 

4349;.  9 

35  00 

108392 

16683.1 

45  00 

142396 

28324.4 

55  00 

178958 

43790.7 

10 

108942 

16848.8 

10 

142982 

28549.0 

10 

179594 

14084  S 

20 

109492 

17015.5 

20 

143569 

28774.8 

20 

180231 

44380.1 

30 

110043 

'17183.1 

30. 

144157 

29001.6 

30 

180809 

44676.7 

40 

!  10595 

17351.6 

40 

144745 

29229.5 

40 

181508 

41974  6 

50 

111147 

175-21.1 

50 

145334 

29158.5 

50 

182147 

45273.8 

36   00 

111699 

17691  .4 

46   00 

145924 

29688.5 

56  CO 

182788 

45574.2 

10 

112252 

17802.7 

10 

146514 

29919.6 

10 

1  S3  430 

45S76.0 

20 

112806 

18034.9 

20 

147105 

30151.8 

20 

1S4073 

46179  0 

30 

1)3360 

18208.1 

30 

147697 

30385.8 

MO 

184717     IK  183.  4 

40 

113915 

18382.1 

40 

148290 

30619.4 

40 

1S53itt    46189.0 

50 

114470 

18557.1 

50 

148883 

30854.8 

50 

\8MQ8 

41  09.-).  9 

37  00 

115025 

18733.1 

47   00 

149478 

31091.4 

•  57   00 

180054 

47404.2 

10 

115582 

18909.9 

10 

150072 

31329.0 

10 

1S7>,02 

47713.8 

20 

116138 

19087.8 

20 

150668 

31567.7 

20 

JS7951 

48024.0 

30 

116096 

19266.5 

30 

151264 

31807.5 

30 

188601 

48336.9 

40 

117254 

19446.3 

40 

151862 

32048.4 

40 

189252 

48650.4 

50 

117812 

19626  9 

50 

152460 

32290.4 

50 

U9905 

48965.3 

38  00 

118371 

19808.5 

48  00 

153058 

32533.  G 

58   00 

190557 

49281.5 

10 

118931 

19991.1 

10 

153658 

32777.8 

10 

191212 

49599.1 

20 

119491 

20174.6 

20 

154258 

33023.1 

20 

191867 

49918.0 

30 

120051 

20359.1 

30 

154859 

33269.6 

30 

192523 

50238.:' 

40 

120613 

20544.5 

40 

155461 

33517.2 

40 

193180 

50559.  N 

50 

121175 

20730  9 

50 

156064 

33765.9 

50 

193839 

50882.8 

39  00 

121737 

20918.2 

49  00 

156667 

34015.8 

59   00 

194498 

51207.1 

10 

122300 

21106.6 

10 

15727! 

34266.7 

10 

195159    51532.9 

20    122864 

21295.8 

20 

157876 

34518.9 

20 

195821    '51859.9 

30    123428 

21480.1 

30 

158482 

34772.1 

30 

1%IS:»   ;5'.'188  4 

40  I1^«9!W 

21677.3 

40 

150089 

35026.5 

40 

197147 

5  25  18.3 

50 

124558 

21869.5 

50 

159690 

85282.01          50 

197812 

52849.5 

40  00 

125124 

22062.7 

60  00 

160304 

35538.71  1   60  00 

198478 

53182.1 

TABLE  VIII.— LENGTHS  OF  ARC  S  CORRESPONDING  TO  ANY  333 
NUMBER  OF  DEGREES,   MINUTES,   OR 
SECONDS  FOR  RADIUS  =  1. 


fo 

Arc  for 

Arc  for 

Arc  for 

«w  0 

Arc  for 

Arc  for 

Arc  for 

Degrees. 

Minutes. 

Seconds. 

c  * 

Degrees. 

Minutes. 

Seconds. 

11 

ll 

1 

.0174533 

.0002909 

.0000048 

31 

.5410521 

.0090175 

.0001503 

2 

.0349086 

.0005818 

.0000097 

32 

.5585054 

.0093084 

.0001551 

3 

.0523599 

.0008727 

.0000145 

33 

.5759587 

.0095993 

.000!  600 

4 

.0698  132 

.0011636 

.0000194 

34 

.5934119 

.0098902 

.0001648 

5 

.OS72M55 

.0014544 

.0000242 

35 

.6108652 

.0101811 

.0001697 

6 

.1047198 

.0017453 

.0000291 

36 

.6283185 

.0104720 

.0001745 

7 

.1221730 

.0020362 

.0000339 

37 

.6457718 

.0107629 

.0001794 

8 

.1396263 

.0023271 

.0000388 

38 

.6632251 

.0110538 

.0001842 

9 

.1570798 

.0026180 

.0000436 

39 

.6806784 

.0113416 

.0001891 

10 

.1745329 

.0029089 

.0000485 

40 

.6981317 

.0116355 

.0001939 

11 

.1919862 

.0031998 

.0000533 

41 

.7155850 

.0119264 

.0001988 

12 

.0034907 

.0000582 

42 

.7330383 

.0122173 

.0002036 

13 

!  2268928 

.0037815' 

.00()i'630 

43 

.7504916 

.0125082 

.0002085 

14 

.2443461 

.0040724 

.0000679 

44 

.7679449 

.0127991 

.0002133 

15 

.2617994 

.0043633 

.0000727 

45 

.7853982 

.0130900 

.0002182 

16 

.2792527 

.0046542 

.f000776 

46 

.8028515 

.0133809 

.  .0002230 

17   .2<)670i;0 

.0049151 

.0000824 

47 

.S203047 

.0136717 

.0002279 

18  1  .3141593 

.005.360 

.0000873 

48 

.8377580 

.0139626 

.0002327- 

19 

.3316126 

.00552t'9 

.0000921 

49 

.8552118 

.0142535 

.0002376 

20 

.3490659 

.0058178 

.0000970 

50 

.8726646 

.0145444 

.0002424 

21 

.3665191 

.0061087 

.0001018 

51 

.8901179 

.0148353 

.0002473 

22 

.3839724 

.0063995 

.0001067 

52 

.9075712 

.0151262 

.0002521 

23 

.4014257 

.0066904 

.0001115 

53 

.9250245 

.0154171 

.0002570 

24 

.4188790  !  .0069K18 

.0001164 

54 

.9424778 

.0157080 

.0002618 

25 

.  4363323 

.0072722 

.0001212 

55 

.9599311 

.0159989 

.0002666 

26 

.4537856 

.0075631 

.0001261 

56 

.9773844 

.0162897 

.0002715 

27 

.4712389 

.0078540 

.0001301 

57 

.9948377 

.0165806 

.0002763 

28 

.4886922 

.0081449 

.0001357 

58 

1.0122910 

.0168715 

.0002812 

29 

.5061455 

.0084358 

.0001406 

59 

1.0297443 

.0171624 

.0002860 

30 

.5235988 

.0087266 

.0001454 

60 

1.0471976 

.0174533 

.0002909 

TABLE  IX.— ACRES  FOR  VARIOUS   LENGTHS  AND  WIDTHS. 


93 

Widths. 

- 

I 

100 

90 

80 

70 

60 

50 

40 

30 

20 

lOol   .229568 

.206612 

.183055 

.160698 

.137741 

.114784 

.091827 

.068871 

.045914 

200 

.459137 

.413223 

.367309 

.321396 

.275482 

.229568 

.183655 

.137741 

.091827 

wo 

.688705 

.619835 

.550964 

.482094 

.413223    .344353 

.275482 

.206612 

.137741 

4(10 

.918274 

.8264J6 

.734619 

.642792 

.550964 

.459137 

.367309 

.275482 

.183655 

500 

1.147842 

1.033058 

.918274 

.803489 

.688705 

.573921 

.459137 

.344353 

.229568 

600 

1.377410 

1.239669 

1.101928 

.9641H7 

.826446 

.688705 

.550964 

.413223 

.275482 

700 

1.606979 

1.446281 

1.285583 

1.124885 

.964187 

.803489 

.642792 

.482094 

.321396 

800 

1.836547 

1.652893 

1.469238 

1.285583 

1.101928 

.918274 

.734619 

.550984 

.367309 

906 

2.066116 

1.859504 

1.652892 

1.446281 

1.239669 

1.033058 

.826446 

.619835 

.413223 

324 


TABLE  X.— TOTAL  GRADES. 


Grade  per  Station. 

J 

00 

0.1 

0.2 

03 

0.4 

0.5 

0.0 

0.7 

0.8 

0.9 

1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

2 

0.2 

0.4 

0.6 

0.8 

1.0 

1.2 

1.4 

1.6 

1.8 

3 

0.3 

0.6 

0.9 

1  .  * 

1.5 

1.8 

2.1 

2.4 

2.7 

4 

0.4 

0.8 

1.2 

1.6 

2.0 

2.4 

2.8 

3kj 
.  •* 

3.6 

5 

0.5 

1.0 

1.5 

2.0 

2.5 

3.0 

3.5 

4.0 

4.5 

6 

0.6 

1.2 

1.8 

2.4 

3.0 

3.6 

4.2 

4.8 

5.4 

y 

0.7 

1.4 

2.1 

2.8 

3.5 

4.2 

4.9 

5.6 

6.3 

8 

0.8 

1.6 

2.4 

3.2 

4.0 

4.8 

5.6 

6.4 

7.2 

9 

0.9 

1.8 

2.7 

3.6 

4.5 

5.4 

6.3 

7.2 

8.1 

10 

1.0 

2.0 

3.0 

4.0 

5.0 

6.0 

7.0 

8.0 

9.0 

11 

1.1 

2/2 

3.3 

4.4 

5.5 

6.6 

7.7 

8.8 

9.9 

12 

1.2 

2.4 

3.6 

4.8 

6.0     j       7.2 

8.4 

9.6 

10.8 

13 

1.3 

2.6 

3.9 

5.2 

6.5 

7.8 

9.1 

10.4 

11.7 

14 

.4 

2.8 

4.2 

5.6 

7.0 

8.4 

9.8 

11.2 

12.6 

15 

.5 

3.0 

4.5 

6.0 

7.5 

9.0 

10.5 

12.0 

13.5 

16 

.6 

3.2 

4.8 

6.4 

8.0 

9.6 

11.2 

12.8 

14.4 

17 

t1 

3.4 

5.1 

6.8 

8.5 

10.2 

11.9 

13.8 

15.3 

18 

!8 

3.6 

5.4 

7.2 

9.0 

10.8 

12.6 

14.4 

16.2 

19 

.9 

3.8 

5.7 

7.6 

9.5 

11.4 

13.3 

15.2 

17.1 

20 

2.0 

4.0 

6.0 

8.0 

10.0 

1-J.O 

14.0 

16.0 

18.0 

21 

2.1 

4.2 

6.3 

8.4 

10.5 

12.6 

14.7 

16.8 

18.9 

22 

2.2 

4.4 

6.6 

8.8 

11.0 

13.2 

15.4 

17.6 

19.8 

23 

2.3 

4.6 

6.9 

9.2 

11.5 

13.8 

16.1 

18.4 

20.7 

24 

2.4- 

4.8 

7.2 

9.6 

12.0 

14.4 

16.8 

19.2 

21.6 

25 

2^5 

5.0 

7.5 

10.0 

12.5 

15.0 

17.5 

20.0 

22.5 

26 

2.6 

5.2 

7.8 

10.4 

13.0 

15.6 

18.2 

20.8 

23.4 

27 

2.7 

5.4 

8.1        10.8 

13.5 

16.2 

18.9 

21.6 

24.3 

28 

2.8 

5.6 

8.4       11.2 

14.0 

IB.  8 

19.6 

22.4 

25  .  2 

29 

2.9 

5.8 

8.7 

11.6 

14.5 

17.4 

20.3 

23  2 

26.1 

30 

3.0 

6.0 

9.0 

12.0 

15.0 

18.0 

21  .0 

24.0 

27.0 

31 

3.1 

6.2 

9.3 

12.4 

15.5 

18.6 

21.7 

24.8 

27.9 

32 

3.2 

6.4 

9.6 

12.8 

16.0 

19.2 

22.4 

25.0 

28.8 

33 

3.3 

6.6 

9.9 

13.2 

16.5 

19.8 

23.1 

26.4 

29.7 

34 

3.4 

6.8       10.2 

13.6 

17.0 

20.4 

23  8 

27.2 

30.6 

35 

3.5 

7.0        10.5 

14.0 

17.5 

21.0 

24.5 

28.0 

31.5 

36 

3.6 

7.2 

10.8 

14.4 

18.0 

21.6 

25.2 

28.8 

32.4 

37 

3.7 

7.4 

11.1 

14.8 

18.5 

22.2 

25.9 

29.6 

33.3 

38 

3.8 

7.6 

11.4 

15.2 

19.0 

22.8 

26.6 

30.4 

34.2 

3D 

3.9 

7.8 

11.7 

15.6 

19  .5 

23.4 

27.3 

31.2 

35.1 

40 

4.0 

8.0 

12.0 

16.0 

20.0 

24.0 

28.0 

32.0 

36.0 

41 

4.1 

8.2 

12.3 

16.4 

20.5 

24.6 

28.7 

32.8 

36.9 

4^* 

4.2 

8.4 

12.6 

16.8 

21.0 

25.2 

29.4 

33.6 

37.8 

43 

4.3 

8.6 

12.9 

17.2 

21.5 

25.8 

30.1 

34.4 

38.7 

44 

4.4 

8.8 

13.2 

17.6 

22.0 

26.4 

30  8 

35.-' 

39.6 

45 

4.5 

9.0 

13.5 

18.0 

22.5 

27.0 

31  5 

36.0 

40.5 

46 

4.6 

9.2 

13.8 

18.4 

23.0 

27.6 

32.2 

36.8 

41.4 

47 

4.7 

9.4 

14.1 

18.8 

23.5 

28.2 

32  9 

37.6 

4-3.3 

48 

4.8 

9.6 

14.4 

19.2 

24.0 

28.8 

33.6 

38.4 

43.2 

49 

49 

9.8 

14.7 

19.6 

24.5 

29.4 

34.3 

39.2 

44.1 

50 

5.0 

10.0 

15.0 

20.0 

25.0 

30.0 

35.0 

40.0 

45.0 

51 

5.1 

10.2 

15.3 

20.4 

25.5 

30.6 

35.7 

40.8 

45.9 

52 

5.2 

10.4 

15.6 

20.8 

26.0 

31.2 

36.4 

41.6 

46.8 

53 

5.3 

10.6 

15.9 

21.2 

26.5 

31.8 

37.1 

4-4.4 

47.7 

54 

5.4 

10.8 

16  2 

21.6 

27.0 

8.'.  4 

37.8 

43.2 

4S.6 

55 

5.5 

11.0 

16.5 

22.0 

27.5 

83.0 

38.5 

44.0 

49.5 

56 

5.6 

11.2 

16.8 

oo  4 

28.0 

33.6 

39.2 

44.8 

50.4 

57 

5.7 

11.4 

17.1 

22.8 

28  .  5 

34.2 

39.9 

45.6 

51  3 

58 

5.8 

11.6 

17.4 

23.2 

29.0 

34,8 

40  (> 

46.4 

53.2 

59 

5.9 

11.8 

17.7 

23.6 

29.5 

35.4 

41.3 

47.'.' 

53.1 

60 

6.0 

12.0 

18.0 

24.0 

30.0 

36.0 

42.0 

48.0 

54.0 

TABLE  XI.— CORRECTION   FOR  THE  EARTH'S  CURVATURE  325 
AND  FOR  REFRACTION. 


Dist. 
D. 

Cor. 
E. 

Dist. 
D. 

Cor. 
E. 

Dist. 
D. 

Cor. 
E. 

Dist. 
D. 

Cor. 
E. 

300 

.00-2 

1800 

.066 

3300 

.223 

4SOO 

.472 

400 

.003 

1900 

.074 

3400 

.237 

4900 

.492 

500 

.005 

2000 

.082 

3500 

.251 

5000 

.512 

GOO 

.007 

2100 

.090 

3600 

.266 

5100 

.533 

700 

.010 

2200 

.099 

3700 

.281 

5200 

.554 

800 

.013 

2300 

.108 

3800 

.296 

1  mile 

.571 

900 

.017 

2100 

.118 

3900 

.312 

2  n  iles 

2.285 

1000 

.020 

2500 

.128 

4000 

.328 

3 

5.142 

1100 

.028 

2600 

.139 

4100 

.345 

4 

9.H2 

1200 

.030 

2700 

.149 

4200 

.362 

5 

14.284 

1300 

.035 

2800 

.161 

4300 

.379 

6 

20.568 

1400 

.040 

2900 

.172 

4400 

.397 

7 

27.996 

1500 

.046 

3000 

.184 

4500 

.415 

8 

36.566 

1600 

.05-2 

3100 

.197 

4600 

.434 

9 

46.279 

1700 

.059 

3200 

.210 

4700 

.453 

10  " 

57.135 

TABLE   XII.-ELEVATION   OF  OUTER  RAIL  ON   CURVES. 


Degree 

Velocity  in  Miles  per  Hour. 

D. 

10 

15 

20 

25 

30 

35 

40 

45 

50 

60 

1 

.006 

.013 

.023 

.036 

.051 

.070 

.091 

.115 

.143 

.205 

2 

.011 

.026 

.046 

.071 

.103 

.140 

.182 

.231 

.285 

.410 

3 

.017 

.038 

.068 

.107 

.154 

.209 

.274 

.346 

.428 

.616 

4 

.023 

.051 

.091 

.142 

.205 

.279 

.365 

.462 

.570 

.821 

5 

.029 

.064 

.114 

.178 

.257 

.349 

.456 

.577 

.712 

6 

.034 

.077 

.137 

.214 

.308 

.419 

.547 

.693 

.855 

7 

.040 

.090 

.160 

.249 

.359 

.489 

.638 

.808 

.997 

8 

.046 

.103 

.182 

.285 

.410 

.559 

.730 

.923 

9 

.051 

.115 

.205 

.321 

.402 

.628 

.821 

10 

.057 

.128 

.228 

.356 

.513 

.698 

.912 

12 

.068 

.153 

.274 

.428 

.616 

.838 

14 

.080 

.180 

.319 

.499 

.718 

.978 

16 

.091 

.£05 

.365 

.570 

.821 

18 

.103 

.231 

.410 

.641 

.923 

20 

.114 

.257 

.456 

.712 

25    * 

.143 

.321 

.570 

.891 

30 

.171 

.385 

.684 

35 

.200 

.449 

.798 

40 

.228 

.513 

.912 

50 

.285 

.641 

326  TABLE  XIII.— COEFFICIENTS  FOR  REDUCING  INCLINED  STADIA 
MEASUREMENTS  TO  THE    HORIZONTAL. 


Inclination. 

0' 

10' 

20' 

30' 

40' 

60' 

0° 

1.00000 

.99999 

.99997 

.99992 

.99986 

.99979 

1 

.99970 

.99959 

.99946 

.99931 

.99915 

.99898 

2 

.99878 

.99857 

.99834 

.99810 

.99784 

.  99756 

3 

.99726 

.99695 

.  99062 

.99627 

.99:>91 

.99553 

4 

.99513 

.99472 

.99429 

.99384 

.99338 

.  99290 

5 

.99240 

.99189 

.99136 

.99081 

.99025 

.98967 

6 

.98907 

.98846 

.98783 

.98718 

.98052 

.98384 

pj- 

.98515 

.98444 

.98371 

.98296 

.98220 

.98142 

8 

.98063 

.97982 

.97899 

.97815 

.97729 

97642 

9 

.97553 

.97162 

.97370 

.97276 

.97180 

.97083 

10 

.96985 

.96884 

.96782 

.96679 

.96574 

.96467 

11 

.96359 

.96249 

.96138 

.96025 

.95911 

.95795 

12 

.95677 

.95558 

.95438 

.95315 

.95192 

.95066 

13 

.94940 

.94881 

.94682 

.94550 

.94417 

.94283 

14 

.94147 

.94010 

.93871 

.93731 

.93589 

.93446 

15 

.93301 

.93155 

.93007 

.92858 

.92708 

.9.  '556 

16 

.92402 

.92247 

.92091 

.91933 

.91774 

.91614 

17 

.91452 

.91288 

.91124 

.90957 

90790 

.90621 

18 

.90451 

.90279 

.90106 

.89932 

.89756 

.89579 

19 

.89400 

.89221 

.89040 

.88857 

.88673 

.88488 

20 

.88302 

.88114 

.87926 

.87735 

.87544 

.87:,51 

21 

.87157 

.86962 

.86765 

.86568 

.86309 

.86168 

22 

.85967 

.85764 

.85560 

.85355 

.85149 

.84941 

23 

.84733 

.84523 

.84312 

.84100 

.83886 

.83672 

24 

.83456 

.83240 

.83022 

.82803 

.82583 

.82301 

25 

.82139 

.81916 

.81691 

.81466 

.81239 

.81011 

26 

.80783 

.80553 

.80322 

.80091 

.79858 

.79624 

27 

.79389 

.79153 

.78916 

.78679 

.78440 

.78200 

28 

.77959 

.77718 

.77475 

.77232 

.76987 

.76742 

29 

.76496 

.76249 

.76001 

.75752 

.75502 

.75251 

30 

.75000 

.74747 

.74494 

.74240 

.73985 

.73730 

81 

.73473 

.73216 

.72958 

.72699 

.72440 

.72179 

32 

.71918 

.71656 

.71394 

.71131 

.70867 

.70602 

33 

.70336 

.70070 

.69804 

.69536 

.69268 

.68999 

34 

.68730 

.68460 

.68189 

.67918 

.67646 

.67374 

35 

.67101 

.66827 

.66553 

.66278 

.66003 

.65727 

36 

.65450 

.65174 

.64896 

.64618 

.64340 

.6-10(51 

37 

.63781 

.63502 

.63221 

.62941 

.62659 

.62378 

38 

.62096 

.61813 

.61530 

.61247 

.60963 

.60679 

39 

.60395 

.60110 

.59825 

.59540 

.59254 

.58968 

40 

.58682 

.58395 

.58108 

.57821 

.57534 

'.57246 

TABLE   XIV.-COEFFICIENTS   FOR   REDUCING  URADIENTER  327 
MEASUREMENTS  TO  THE  HORIZONTAL. 


Inclination. 

0' 

10' 

20' 

30' 

40' 

60' 

0° 

100.00 

99.99 

99.99 

99.98 

99.97 

99.96 

1 

99.95 

99.94 

99.92 

99.91 

99.89 

99.87 

2 

99.84 

99.82 

99.79 

99.77 

99.74 

99.71 

3 

99.67 

99.64 

99.60 

99.57 

99.53 

99.49 

4 

99.45 

99.40 

99.35 

99.31 

99.26 

99.21 

5 

99.15 

99.10 

99.04 

98.98 

98.93 

98  87 

6 

98.80 

98.74 

98.67 

98.61 

98.54 

98.47 

7 

98.39 

98.32 

98.24 

98.17 

98.09 

98.01 

8 

97.93 

97.84 

97.76 

97.66 

97.58 

97.49 

9 

97.40 

97.31 

97.21 

97.11 

97.02 

96.91 

10 

96.81 

96.71 

96.60 

96.50 

96.39 

96.28 

11 

96.17 

96.06 

95.94 

95.83 

95.71 

95.59 

12 

95.47 

95.35 

95.23 

95.10 

94.97 

94.85 

13 

94.72 

94.59 

94.46 

94.3-2 

94.18 

94.05 

14 

93.91 

93.78 

93.63 

93.49 

93.31 

93.20 

15 

93.05 

92.90 

92.75 

92.60 

92.45 

92.29 

16 

92.14 

91.98 

91.82 

9J.66 

91.50 

91.34 

17 

91.18 

91.01 

90.84 

90.67 

90.50 

90.33 

18 

90.16 

89.98 

89.81 

89.63 

89.45 

89  27 

19 

89.09 

88.91 

88.73 

88.54 

88.36 

88.17 

20 

87.98 

87.79 

87.60 

87.41 

87.21 

87.01 

21 

86.82 

86.63 

86.43 

86.23 

86.02 

85.82 

23 

85.62 

85.42 

85.21 

85.00 

84.79 

84.58 

23 

84.38 

84.16 

83.95 

83.73 

83.52 

83.30 

24 

83.08 

82.87 

82.65 

82.43 

82.20 

81.98 

25 

81.76 

81.53 

81.31 

81.08 

80.85 

80.62 

26 

80.  30 

80.16 

79.93 

79.69 

79.45 

79.22 

27 

78.99 

78.75 

78.51 

78.27 

78.03 

77.79 

28 

77.54 

77.30 

77.06 

76.81 

76.57 

76.32 

29 

76.07 

75.82 

75.57 

75.32 

75.07 

74.82 

30 

74.57 

74.31 

74.06 

73.80 

73.55 

73.29 

328  TABLE  XV.— OFFSETS  FOR  TRANSITION  CURVES. 

s       o 

AK=t  =       8iH=.    Fi>?.  130. 

' 


Deg.  of 
Offset 

Rad    of 
Offset 

Lengths  of  Transition  Curves. 

Curve 

Curve 

100 

150 

200 

240 

250 

300 

320 

360 

400 

0°  10' 

34377 

.01 

.03 

.05 

.07 

.08 

.11 

.12 

.16 

.19 

20 

17189 

.02 

.05 

.10 

.14 

.15 

.81 

.:>/, 

30 

11459 

.04 

.08 

.15 

.21 

22 

.'33 

.37 

.47 

.5s 

40 

8595  4 

.05 

.11 

.19 

.28 

iao 

.44 

.50 

.63 

.78 

50 

6875.5 

.06 

.14 

.24 

.35 

.38 

.55 

.62 

.79 

.97 

1     00 

5729.6 

.07 

.16 

.29 

.42 

.46 

.65 

.74 

.94 

1.1(5 

10 

4911   1 

.08 

,19 

.34 

.49 

.53 

.76 

.8? 

1.10 

1.36 

20 

4297.2 

.10 

22 

.39 

.56 

.61 

.87 

.!)'.) 

1  .  26 

1.55 

30 

3819.7 

.11 

!25 

.44 

.63 

.68 

.98 

1.12 

1.41 

1  .  75 

40 

3437.8 

.12 

.27 

.48 

.70 

.76 

1.09 

1.24 

1.57 

1.94 

50 

3135.2 

.13 

.30 

.53 

.  77 

.83 

1.20 

1.37 

1  .73 

2.13 

2    00 

2S64.8 

.15 

.33 

.58 

.84 

.91 

1.31 

1.49 

1.83 

2.33 

10 

2644.4 

.16 

.35 

.63 

.91 

.98 

1.42 

1.61 

2.01 

2.52 

20 

2455.5 

.17 

.38 

.68 

.98 

.06 

1.53 

f,78 

2.^0 

0     ij- 

30 

2291.8 

.18 

.41 

.73 

.  05 

.14 

1.64 

1.86 

2.36 

2^91 

40 

2148.6 

.19 

.44 

.78 

.12 

.21 

1.74 

1.99 

2.51 

3.10 

50 

2022.2 

.21 

.46 

.82 

.19 

.29 

1.85 

2.11 

X.'  67 

3.30 

3    00 

1909.9 

.22 

.49 

.87 

.26 

.36 

1.96 

2.23 

2.  S3 

3.49 

10 

1809.3 

.23 

.5-2 

.92 

.33 

.44 

2.07 

2.86 

2  .  98 

3.68 

20 

1718.9 

.24 

.55 

.97 

.40 

.51 

2.18 

2.  48 

3.14 

3.88 

30 

1637.0 

.25 

.57 

1.02 

.47 

.59 

2.29 

2.61 

3.30 

4.07 

40 

1562.6 

.27 

.60 

1.07 

.54 

.67 

2.40 

2.73 

3.45 

4.26 

50 

1494.7 

.28 

.63 

1.11 

.61 

.74 

2.51 

2.85 

8.  (51 

4.46 

4    00 

1432.4 

.29 

.65 

.16 

1.68        .82 

2.62 

2.98 

3.77 

4.65 

10 

1375.1 

.30 

.68 

.21 

1.74        .89 

2.73 

3.10 

3.92 

4.S4 

20 

13-^.2 

.32 

.71 

.26 

1.81        .97 

2.83 

3.23 

4.08 

5.04 

30 

1273.2 

.33 

.74 

.31 

1.88        .04 

2.94 

3.35 

4.24 

5.23 

40 

1227.8 

.34 

.76 

.36 

1.95        .12 

3.05 

3.47 

4.39 

5.48 

50 

1185.4 

.35 

.79 

.41 

2.02 

.20 

3.16 

3.  GO 

4.55 

5.62 

5    00 

1145.9 

.36 

.82 

.45 

2.09 

2.27 

3.27 

3.72 

4.71 

5.81 

10 

1109.0 

.38 

.85 

.50 

2.16 

2.35 

3.38 

3.84 

4.86 

6.00 

20 

1074.3 

.39 

.87 

.55 

2.23 

2.42 

3.49 

3.97 

5.02 

6.20! 

30 

1041.7 

.40 

.90 

.60 

2.30 

2.49 

3.60 

4.09 

5.18 

6.39! 

40 

1011.1 

.41 

.93 

.65 

2.37 

2.57 

3.71 

4.22 

5.33 

6.58 

50 

982.2 

.42 

.95 

.70 

2.44 

2.65 

3.81 

4.34 

5  49 

6.78 

6    00 

954.9 

.44 

.98 

.74 

2.51 

2.73 

3.92 

4.46 

5.65 

6.97 

10 

929.1 

.45 

1.01 

79 

2.58  '  2.80 

4.03 

4.59 

5.80 

7.16 

20 

904.7 

.46 

1.04 

.84 

2.65  <  2.88 

4.14 

4.71 

fi  .  96 

7.36 

30 

881.5 

.47 

1.06 

.89 

2.72      2.95 

4.25 

4.83 

6.12 

7.55 

40 

859.4 

.48 

1.09 

.94 

2.79      3.03 

4.36 

4.96 

6.27 

7.74 

50 

838.5 

.50 

1.12 

.99 

2.86      3.10 

4.47 

5.08 

6.43 

7.93 

7    00 

818.5 

.51 

.14 

2.03 

2.93      3.18 

4.1$ 

5.20 

G  .  58 

8.12 

10 

799.5 

.52 

.17 

2.08 

3.00      3.25 

4.68 

5.33 

6.74 

8.32 

20 

781.3 

.53 

.20 

2.13 

3.07      3.33 

4.79 

5.45 

6.SK) 

8.51 

30 

763.9 

.55 

1.23 

2.18 

3.14      3.41 

4.90 

5.57 

7.05 

8.70 

40 

747.3 

.56 

.25 

2.23 

3.21      3.48 

5.01 

5.70 

7.21 

8.89 

50 

731.4 

.57 

.28 

2.28 

3.28      3.56 

5.12 

5.82 

7.36 

9.09 

8    00 

716.2 

.58 

.31 

2.33 

3.35      3.63 

5  23 

5.95 

7  .  5  '2 

9  28 

10 

701.6 

.59 

.34 

2.37 

3.42      3.71 

5.33 

6.07 

7.68 

9.47 

20 

687.5 

.61 

.36 

2.42 

3  49 

3.78 

5.44 

6.19 

7.83 

9.66 

30 

674.1 

.62 

1.39 

2.47 

3.56 

3.86 

5.55 

6.31 

7.99 

9.85 

40 

661.1 

.63 

.42 

2.52 

3.63      3.93 

5.66 

6.44 

8.14 

10.05 

50 

648.6 

.64 

.44 

2.57 

8.69  ;  4.01 

5  .  77' 

0.56 

8.  80 

10.24 

9    00 

636.6 

.65 

.47 

2.62 

3.76  j  4.08 

5.88 

6.68 

8.45 

10.43 

10 

625.0 

.67 

.50 

2.66 

383      4.16 

5.99 

6.  SI 

8.61 

10.62 

20 

613.9 

.68 

.53      2.71 

3.90      4.23 

6.09 

(5.93 

8.76 

10.81 

30 

603.1 

.69 

.55      2.76 

3.97  I  4.31 

6.20 

7.05 

8.92 

11.  i'(i 

40 

592.7 

.70 

.58      2.81 

4.04      4.39 

6.31 

7.18 

9.08 

11.19 

50 

582.7 

.71 

.61      2.86 

4.11 

4.46 

6.42 

7.30 

9.23 

11.39 

10    00 

573.0 

.73 

1.64 

2.91 

4.18 

4.54 

6.53 

7.42 

9.39 

11.58 

TABLE  XV. -OFFSETS  FOR  TRANSITION  CURVES. 


329 


Deg.  of 
Offset 
Curve. 

Had.  ol 
Offsei 

Curve. 

Lengths  of  Transition  Curves. 

100 

150      200 

240 

250 

300 

320 

360 

400 

10°  10' 

563.0 

.74 

.66      2.95 

4.25 

4.61 

6.63 

7.55 

9.54 

11.77 

20 

554.5 

.75 

.69      3.00 

4.32 

4.69 

6.74 

7.67 

9.70 

11.96 

30 

546.7 

.16 

.72  i  3.05 

4.39 

4.76 

6.85 

7.79 

9.85 

12.15 

40 

537.1 

.78 

.74 

3.10 

4.46 

4.84 

6.96 

7.91 

10.  0: 

12.34 

50 

528.9 

.79 

.77 

3.15 

4.53 

4.91 

7.07 

8.03 

10.16 

12.53 

11     00 

520.9 

.80 

.80 

3.19 

4.60 

4.99 

7.17 

8.16 

10.3', 

12.7 

10 

513.1 

.81 

.83 

3  24 

4.67 

5.06 

7.28 

8.28 

10.47 

12.91 

20 

505.6 

.82 

.86 

3.29 

4.74 

5.14 

7.39 

8.40 

10.6:: 

13.10 

30 

498.2 

.84 

.88 

3.34 

4.81 

5.21 

7.50 

8.53 

10.78 

13.29 

40 

491.1 

.85 

.91 

3.39 

4.87 

5.29 

.61 

8.65 

10.93 

13.48 

50 

484.2 

.86 

.93 

3.44 

4.94 

5.36 

.71 

8.77 

11.09 

13.67 

12     00 

477.5 

.87 

.96 

3.48 

5.01 

5.44 

.82 

8.89 

11.24 

13.86 

10 

470.9 

.88 

.99 

3.53 

5.  OS 

5.51 

.93 

9.02 

11.40 

14.05 

20 

464.6 

.90 

2.02 

3.58 

5.15 

5.59 

8.04 

9.14 

11.66 

14.24 

30 

458.4 

.91 

2.04 

3.63 

5.22 

5.66 

8.14 

9.26 

11.71 

14.43 

40 

45-2.3 

.92 

2.07 

3.68 

5.29 

5.74 

8.25 

9.38 

11.86 

14.62 

50 

446.5 

.93 

2.10 

3.73 

5.36 

5.81 

8.36 

9.51 

12.01 

14.81 

13    00 

440.7 

.94 

2.12 

3.77 

5.43 

5.89 

8.47 

9.63 

12.17 

15.00 

10 

435.2 

.96 

2.15 

3.82 

5.50 

5.96 

8.58 

9.75 

12.32 

15.19 

20 

429.7 

.97 

2.18 

3.87 

5,57 

6.04 

8.68 

9.87 

12.47 

15.37 

30 

424.4 

.98 

2.21 

3  92 

5.61 

6J1 

8.79 

9.99 

12.63 

15.56 

40 

419.2 

.99 

2.23 

3.97 

5.71 

6.19 

8.90 

10.12 

12.78 

15.75 

50 

414.2 

1.01 

2.26 

4.01 

5.77 

6.26 

9.00 

10.23 

12.94 

15.94 

14    00 

409.3 

.02 

2  29 

4.06 

5.84 

6.34 

9.11 

10.36 

13.09 

16.13 

10 

404.4 

.03 

2^31 

4.11 

5.91 

6.41 

9.22 

10.48 

13.24 

16.32 

20 

399.7 

.04 

2.34 

4.16 

5.98 

6.49 

9.33 

10.60 

13.40 

16.50 

30 

395.1 

.05 

2.37 

4.21 

6.05 

6.56 

9.43 

10.72 

13.55 

16.69 

40 

390.7 

.07 

2.40 

4.25 

6.12 

6.64 

9.54 

10.85 

13.70 

16.88 

50 

388.3 

.08 

2.42 

4.30 

6.19 

6.71 

9.65 

10.97 

13.85 

17.07 

15    00 

382.0 

.09 

2.45 

4.35 

6.26 

6.79 

9.75 

11.09 

14  01 

17.25 

10 

377.8 

.10 

2.48 

4.40 

6.33 

6.86 

9.86 

11.21 

14.16 

17.44 

20 

373.7 

.11 

2.50 

4.45 

6.40 

6.94 

9.97 

11.33 

14.31 

17.63 

:30 

369.7 

.13 

2.53 

4.50 

6.46 

".01 

10.08 

11.45 

14.46 

17.82 

40 

365.7 

.14 

2.56 

4.54 

6.53 

/.09 

10.18 

11.57 

14.62 

18.00 

50 

361.9 

.15 

2.59 

4.59 

6.60 

7.16 

10.29 

11.69 

14.77 

18.19 

16    00 

358.1 

.16 

2.61 

4.64 

6.67 

^.24 

10.40 

11.82 

14.92 

18.38 

10 

354.4 

.17 

2.64 

4.69 

6.74 

".31 

10.50 

11.94 

15.07 

18.56 

20 

350.8 

.19 

2.67 

4.74 

6  81 

'.38 

10.61 

12.06 

15.23 

18.75 

30 

347.2 

.20 

2.69 

4.78 

6.88 

".48 

10.72 

12.18 

15.38 

18.93 

40 

343.8 

.21 

2.72 

4.83 

6.95      ".53 

10.82 

12.30 

15.53 

19.12 

50 

340.4 

.22 

2.75 

4.88 

.01 

".61 

10.93 

12.42 

15.68 

19.31 

17    00 

337.0 

.24 

2.78 

4.93 

.08 

^.68 

11.03 

12.54 

15.83 

19.49 

10 

333.8 

.25 

2.80 

4.98 

.15 

".76 

11.14 

12.66 

15.981  19.68 

20 

330.6 

.26 

2.83 

5.02 

.22 

".83 

11.25 

12.78 

16.14    19.86 

30 

327.4 

27 

2.86 

5.07 

.29 

".91 

11.35 

12.90 

16.29!  20.05 

40 

324.3 

^28 

2.88 

5.18 

.36 

".98 

11.46 

13.02 

16.441  20.23 

50 

321.3 

.30 

2.91 

5.17 

.43 

8.05 

11.57 

13.14 

16.59    20.42 

18    00 

318.3 

.31 

2.94 

5.21 

.50 

8.13 

11.67 

13.26 

16.74    20.60 

10 

315.4 

.32 

2.97 

5.26 

.51) 

8.20 

11.78 

13.38 

16.89    20.79 

20 

312.5 

.33 

2.99 

5.31 

.63 

8.28 

11.88 

13.50 

17.04    20.97 

30 

309.7 

.34 

3.02 

5.36 

.70 

8.35 

11.99 

13.62 

17.19    21.15 

40 

306.9 

.36 

3.05 

5.41 

77 

8.43 

12  10 

13.74 

17.34    21.34 

50 

304.2 

.37 

3.07 

5.45 

'.84 

8.50 

12.20 

13.86 

17.49    21.52 

19    00 

301.6 

.38 

3.10 

5.50 

.91 

8.57 

12.31 

13.98 

17.64    21.70 

10 

298.9 

.39 

3.13 

5.55 

.97 

8.65 

12.41 

14  10 

17.79    21.89 

20 

296.4 

.40 

3.16 

5.60 

8.04 

8.72 

12.52 

14.22 

17.941  22.07 

30 

293.8 

.42 

3.18 

5.65 

8.11 

8.80 

12.62 

14.34 

18.09    22.25 

40 

291.3 

.43 

3.21 

5  69 

8.18 

8.87 

12.73 

14.46 

18.24    22.44 

50 

288.9 

.44 

3  .  24 

5.74 

8.25 

8.94 

12.84 

14.58 

18  39    22.62 

20    00 

286.5 

1.45 

3.26 

5.79 

8.32 

9.02 

12.94 

14.70 

18.54    22.80 

330  TABLE  XVI.— TRANSITION  CURVES. 

Table  gives    d  =  ^  vers  ^.    Then  AO  =  f-  -  d. 


Degree 

Lengths  of  Transition  Curves. 

of  Offset 
Curve. 

100 

150       200 

240 

250       300 

320 

360 

400 

0°  10' 

20 

.001 

.001 

30 

.001 

.001 

.001 

.002 

40 

.001 

.001 

.001 

.002 

.002 

.003 

50 

.001 

.001 

.001 

.002 

.003 

.004 

.005 

1    00 

.001 

.002 

.002 

.003 

.004 

.006 

.008 

10 

.001 

.001 

.002 

.003 

.005 

.005 

.008 

.01 

20 

.001 

.002 

.003 

.003 

.006 

.007 

.01 

.01 

30 

.001 

.002 

.004 

.004 

.007 

.009 

.01 

.02 

40 

.001 

.003 

.005 

.005 

.009 

.01 

.02 

.02 

50 

.001 

.003 

.006 

•006 

.01 

.01 

.02 

.03 

2    00 

.002 

.004 

.007 

.007 

.01 

.02 

.02 

.03 

10 

.001 

.002 

.005 

.008 

.009 

.02 

.02 

.03 

.04 

20 

.001 

.002 

.005 

.009 

.01 

.02 

.02 

.03 

.04 

30 

.001 

.002 

.006 

.01 

.01 

.02 

.02 

.03 

.05 

40 

.001 

.003 

.007 

.01 

.01 

.02 

.03 

.04 

.05 

50 

.001 

.003 

.008 

.01 

.01 

.03 

.03 

.04 

.06 

3    00 

.001  1     .004 

.009 

.01 

.02 

.03 

.04 

.05 

.07 

10 

.001 

.004 

.01 

.02 

.02 

.03 

.04 

.06 

.08 

20 

.001 

.004 

.01 

.02 

.02 

.04 

.04 

.06 

.08 

30 

.001 

.005 

.01 

.02 

.02 

.04 

.05 

.07 

.09 

40 

.002 

.005 

.01 

.02 

.03 

.04 

.05 

.07 

.10 

50 

.002 

.006 

.01 

.02 

.03 

.05 

.06 

.08 

.11 

4    00 

.002 

.006 

.02 

.03 

.03 

.05 

.06 

.09 

.12 

10 

.002 

.007 

.02 

.03 

.03 

.06 

.07 

.10 

.13 

20 

.002 

.008 

.02 

.03 

.03 

.06 

.07 

.10 

.14 

30 

.002  !     .008 

.02 

.03 

.04 

.07 

.08 

.11 

.15 

40 

.003 

.009 

.02 

.04 

.04 

.07 

.08 

.12 

.17 

50 

.003 

.009 

.02 

.04 

.04 

.08 

.09 

.13 

.18 

5    00 

.003 

.01 

.02 

.04 

.05 

.08 

.10 

.14 

.19 

10 

.003 

.01 

.03 

.04 

.05 

.09 

.10 

.15 

.20 

20 

.003 

.01 

.03 

.05 

.05 

.09 

.11 

.16 

.22 

30 

.004        .01 

.03 

.05 

.06 

.10 

.12 

.17 

.23 

40 

.004 

.01 

.03 

.05 

.06 

.10 

.13 

.18 

.24 

50 

.004 

.01 

.03 

.06 

.06 

.11 

.13 

.19 

.26 

6    00 

.004 

.01 

.03 

.06 

.07 

.12 

.14 

.20 

.27 

10 

.005 

.02 

.01 

.06 

.07 

.12 

.15 

.21 

.29 

20 

.005 

.02 

.04 

.07 

.07 

.13 

.16 

.22 

.31 

30 

.005 

.02 

.04 

.07 

.08 

.14 

.16 

.23 

.32 

40 

.005 

.02 

.04 

.07 

.08 

.14 

.17 

.25 

.34 

50 

.006 

.02 

.04 

.08 

.09 

.15 

.18 

.26 

.36 

7     00 

.006 

.02 

.05 

.08 

.09 

.16 

.19 

.27 

.37 

10 

.006 

.02 

.05 

.08 

.09 

.16 

.20 

.28 

.39 

20 

.006 

.02 

.05 

.09 

.10 

.17 

.21 

.30 

.41 

30 

.007 

.02 

.05 

.09 

.10 

.18 

.22 

.31 

.43 

40 

.007 

.02 

.06 

.10 

.11 

.19 

.23 

.33 

.45 

50 

.007 

.02 

.06 

.10 

.11 

.20 

.24 

.34 

.47 

8    00 

.008  !     .03 

.06 

.11 

.12 

.21 

.25 

.86 

.49 

10 

.008  i     .03 

.06 

.11 

.12 

.21 

.26 

.37 

.51 

20 

.008        .03 

.07 

.11 

.13 

.22 

.27 

.39 

.53 

30 

.009  1     .03 

.07 

.12 

.13 

.23 

.28 

.40 

.55 

40 

.009 

.03 

.07 

.12 

.14 

.24 

.29 

.42 

.57 

50 

.009 

.03 

.07 

.13 

.15 

.25 

.30 

.43 

.59 

9    00 

.01 

.03 

.08 

.13 

.15 

.26 

.32 

.45 

.62 

10 

.01 

.03 

.08 

.14 

.16 

.27 

.33 

.47 

.64 

20 

.01 

.03 

.08 

.14 

.16 

.28 

.34 

.48 

.66 

30 

.01 

.04 

.09. 

.15 

.17 

.29 

.35 

.50 

.69 

40 

.01 

.04 

.09 

.15 

.17 

.30 

.36 

.52 

.71 

50 

.01 

.04 

.09 

.16 

.18 

.31 

.38 

.54 

.74 

10    00 

.01 

.04 

.10 

.16 

.19 

.32 

.39 

.55 

.76 

TABLE  xvi.— TRANSITION  CURVES. 


331 


Degree 

Lengths  of  Transition  Curves. 

of  Offset 

Curve. 

100 

150 

200 

240 

250 

300 

320 

3(50 

400 

10°  10' 

.01 

.04 

.10 

.17 

.19 

.33 

.40 

.57 

.79 

20 

.01 

.04 

.10 

.18 

.20 

.34 

.42 

.59 

.81 

30 

.01 

.04 

.10 

.18 

.20 

.35 

'.43 

.61 

.84 

40 

.01 

.05 

.11 

.19 

.21 

.37 

.44 

.63 

.87 

50 

.01 

.05 

.11 

.19 

.22 

.38 

.46 

.65 

.89 

11     00 

.01 

.05 

.12 

.20 

.23 

.39 

.47 

.67 

.92 

10 

.01 

.05 

.12 

.21 

.23 

.40 

.49 

.69 

.95 

20 

.02 

.05 

.12 

.21 

.24 

.41 

.50 

.71 

.98 

30 

.02 

.05 

.13 

.22 

.25 

.42 

.52 

.73 

.01 

40 

.02 

.05 

.13 

.22 

.25 

.44 

.53 

.76 

.04 

50 

.02 

.06 

.13     j     .23 

.26 

.45 

.55 

.78 

.07 

12    00 

.02 

.06 

.14 

.24 

.27 

.46 

.56 

.FO 

.10 

10 

.02 

.06 

.14 

.24 

.27 

.48 

.58 

.82 

.13 

20 

.02 

.06 

.14 

25 

.28 

.49 

.59' 

.84 

.16 

30 

.02 

.06 

.15 

^26 

.29 

.50 

.61 

.87 

.19 

40 

.02 

.06 

.15 

.26 

.30 

.52 

.63 

.89 

.22 

50 

.02 

.07 

.16 

.27 

.31 

.53 

.64 

.91 

.25 

13    00 

.02 

.07 

.16 

.28 

.32 

.54 

.66 

.94 

.29 

10 

.02 

.07 

.16 

.-.'9 

.32 

.55 

.68 

.96 

.32 

30 

.02  * 

.07 

.17 

.29 

.33 

.57 

.69 

.99 

.35 

30 

.02 

.07 

.17 

.30 

.34 

.59 

.71 

.01 

.39 

40 

.02 

.07 

.18 

.31 

.35 

.60 

.73 

.04 

.42 

50 

.02 

.08 

.18 

.31 

.36 

.61 

.75 

.06 

.46 

U    00 

.02 

.08 

.19 

.32 

.36 

.62 

.76 

.09 

.49 

10 

.02 

.08 

.19 

.33 

.37     i       .64 

.78 

.11 

.53 

20 

.02 

.08 

.20 

.34 

.38 

.66 

.80 

.14 

.56 

::o 

.03 

.08 

.20 

.35 

.39 

.68 

.82 

.17 

.60 

40 

.03 

.09 

.20 

.35 

.40 

.69 

.84 

.19 

.64 

50 

.03 

.09 

.21 

.36 

.41 

.71 

.86 

.22 

.67 

15    00 

.03 

.09 

.21 

.37 

.42 

.72 

.88 

.25 

.71 

10 

.03 

.09 

.22 

38 

'.43 

.74 

.90 

.28 

.75 

20 

.03 

.09 

.22 

.39 

.44 

.75 

.92 

.30 

.79 

30 

.03 

.10 

.23 

.39 

.45 

77 

.94 

.33 

.83 

40 

.03 

.10 

.23 

.40 

.46 

'.79 

.96 

.36 

.87 

50 

.03 

.10 

.24 

.41 

.47 

.80 

.98 

.39 

.91 

16    00 

.03 

.10 

.24 

.42 

.48 

.82 

1.00 

.42 

1.95 

10 

.03 

.10 

.25 

.43 

.49 

.84 

1.02 

.45 

1.99 

20 

.03 

.11 

.2') 

.44 

.50 

.86 

.04 

.48 

2.03 

80 

.03 

.11 

.26 

.45 

.51 

.87 

.06 

.51 

2.07 

40 

.03 

.11 

.26 

.46 

.52 

.89 

.08 

.54 

2.11 

50 

.03 

.11 

.27 

.47 

.53 

.91 

.10 

.57 

2.15 

17     00 

.03 

.12 

.27 

.48 

.54 

.93 

.13 

.60 

2.20 

10 

.04 

.12 

.28 

.48 

.55 

.95 

.15 

.63 

2.24 

20 

.04 

.12 

.29 

.49 

.56 

.96 

.17 

.66 

2.28 

30 

.04 

.12 

.29 

.50 

.57 

.98 

.19 

.70 

2.33 

40 

.04 

J3 

.30 

.51 

.58 

1.00 

.22 

.73 

2.37 

50 

.04 

.13 

.30 

.52 

.59 

1.02 

.24 

.76 

-2.42 

18    00 

.04 

.13 

.31 

.53 

.60 

1.04 

.26 

.80 

2.46 

10 

.04 

.13 

.*! 

.54 

.61 

1.06 

.29 

.83 

2.51 

20 

.04 

.13 

.32 

.55 

.62 

.08 

!ai 

.86 

2.55 

30 

.04 

.14 

.33 

.56 

.64 

.10 

.33 

.90 

2.60 

40 

.04 

.14 

.33 

.57 

.65 

.12 

.36 

.93 

2.65 

50 

.04 

.14 

.34 

.58 

.66 

.14 

.38 

.96 

2.70 

19    00 

.04 

.14 

.34 

.59 

.67 

.16 

.41 

2.00 

2.74 

10 

.04 

.15 

.35 

.60 

.68 

.18 

.43 

2.04 

2.79 

20 

.04 

.15 

.36 

.61 

.69 

.20 

.46 

2.07 

2.84 

30 

.05 

.15 

.36 

.63 

.71 

.22 

.48 

2.11 

2.89 

40 

.05 

.16 

.37 

.64 

72  • 

1.24 

.51 

2.14 

2.94 

50 

.05 

.16 

.37 

.65 

•78 

.26 

.53 

2.18 

2.99 

20    00 

.05 

.16 

.38 

.66 

.74 

1.28 

1.56 

2.22 

3.04 

B32  TABLE  XVII.-DEFLECTION  ANGLES  FOR  TRANSITION  CURVES, 

TO  TENTHS  OU%  A  MINUTE,   FOR  CHORDS   OF   1  TO   5  CHAINS, 

IN   TERMS   OF   ANGLE  FOR   CHORD   OF   1    CHAIN. 


1 

2 

g 

4 

•°i 

1 

a 

I 

4 

5 

O'.l 

0'.4 

0'.9 

i'.a 

2'.  5 

6'.1 

24'.  4 

54'.  9 

1°  38' 

2°  32' 

0  .2 

0  .8 

1  .8 

3  .2 

5  .0 

G  .2 

24  .8 

55  .8 

39 

35 

0  .8 

1  .2 

2  .7 

4  .8 

7  .5 

G  .3 

25  .2 

56  .7 

41 

37 

0  .4 

1  .0 

3  .G 

G  .4 

10  .0 

6  .4 

25  .6 

57  .6 

42 

40 

0  .5 

2  .0 

4  .5 

8  .0 

12  .5 

6  .5 

26  .0 

58  .5 

44 

42 

0  .0 

2  .4 

5  .4 

9  .(i 

15  0 

6  .'i 

26  .4 

59  .4 

46 

45 

0  .7 

2  .8 

G  .3 

11  .2 

17  .5 

G  .7 

26  .8 

1°   0 

47 

47 

0  .8 

3  .2 

7  .2 

1.'  .8 

20  .0 

6  .8 

27  .2 

1 

49 

50 

0  .9 

3  .6 

8  .1 

14  .4 

22  .5 

6  .9 

v'7  .6 

B 

50 

52 

1  .0 

4  .0 

9  .0 

10  .0 

25  .0 

.0 

•Jd  .0 

3 

52 

55 

1  .1 

4  .4 

9  .9 

17  .G 

27  .5 

.1 

28  .4 

4 

54 

57 

i  .2 

4  .8 

10  .8 

19  .2 

30  .0 

0 

28  .8 

5 

55 

3   0 

1  .3 

5  .2 

11  .7 

20  .8 

3-J  .5 

!a 

•29  .2 

G 

57 

2 

1  .4 

5  .6 

12  .G 

2-2  .4 

35  .0 

.4 

29  .6 

7 

58 

5 

1  .5 

6  .0 

13  .5 

24  .0 

37  .5 

.5 

30  .0 

8 

•-'   0 

7 

1  .6 

G  .4 

14  .4 

25  .G 

40  .0 

.6 

30  .4 

8 

2 

10 

1  .7 

0  .8 

15  .3 

27  .2 

42  .5 

r- 

30  .8 

5) 

3 

12 

1  .8 

7  2 

1G  .2 

28  .8 

45  .0 

7  !B 

31  .2 

10 

5 

15 

1  .9 

7  !e 

17  .1 

30  .4 

47  .5 

7  .9 

31  .6 

Ik 

G 

17 

2  .0 

8  .0 

18  .0 

32  .0 

50  .0 

8  .0 

32  .0 

12 

8 

20 

2  .1 

8  .4 

18  .9 

33  .G 

52  .5 

8  .1 

3-2  .4 

13 

10 

22 

8  .8 

19  .8 

35  .2 

55  .0 

8  .2 

32  .8 

14 

11 

25 

2  !a 

9  .2 

i.0  .7 

36  .8 

57  .5 

8  .3 

38  .2 

15 

13 

27 

2  .4 

9  .G 

21  .6 

36  .4 

1°  0' 

8  .4 

33  .6 

1G 

14 

30 

2  .5 

10  .0 

S2  .5 

40  .0 

2 

8  .5 

34  .0 

17 

16 

32 

2  .6 

10  .4 

*3  .4 

41  .6 

5 

8  .6 

34  .4 

17 

18 

35 

2  7 

10  .8 

21  .3 

4.J  .4 

7 

8  .7 

34  .8 

18 

19 

37 

2  is 

11  .2 

25  .2 

44  .8 

10 

8  .8 

35  .2 

19 

21 

40 

2  .9 

11  .6 

26  .1 

46  .4 

12 

8  .9 

35  .6 

20 

22 

42 

3  .0 

12  .0 

27  .0 

48  .0 

15 

9  .0 

26  .0 

21 

24 

45 

3  .1 

12  .4 

27  .9 

49  .6 

17 

9  .1 

3(5  .4 

22 

26 

47 

3  .2 

12  .8 

28  .8 

51  .2 

20 

9  .2 

815  .8 

23 

27 

50 

3  .3 

13  .2 

29  .7 

52  .8 

22 

9  .8 

3;  .2 

24 

29 

52 

3  .4 

13  .G 

80  .6 

54  .4 

25 

9  .4 

37  .6 

25 

30 

55 

3  .5 

14  .0 

3!  .5 

56  .0 

er 

9  .5 

38  .0 

26 

32 

57 

3  .6 

14  .4 

32  .4 

57  .6 

so 

9  .6 

38  .4 

26 

34 

4   0 

3  .7 

14  .8 

33  .3 

59  .2 

32 

9  .7 

38  .8 

27 

35 

2 

3  .8 

15  .2 

34  2 

1°  1' 

35 

9  .8 

39  .2 

28 

37 

5 

3  .9 

15  .G 

35  .1 

2 

37 

9  .9 

39  .6 

n 

38 

7 

4  .0 

1G  .0 

36  .0 

4 

40 

10  .0 

40  .0 

30 

40 

10 

4  .1 

16  .4 

36  .9 

6 

42 

10  .1 

40  .4 

31 

42 

12 

4  .•„» 

16  .8 

37  .8 

45 

10  .2 

40  .8 

32 

43 

15 

4  .3 

17  .2 

38  .7 

9 

47 

10  .3 

41  .2 

88 

45 

17 

4  .4 

17  .G 

3D  .6 

10 

50 

10  4 

41  .6 

34 

46 

20 

4  .5 

18  .0 

40  .5 

12 

52 

10  .5 

42  .0 

35 

48 

22 

4  .0 

18  .4 

41  .4 

14 

55 

10  .6 

42  .4 

35 

50 

25 

4  -.7 

18  .8 

4-2  .3 

15 

57 

10  .7 

42  .8 

* 

51 

27 

4  .8 

19  .2 

43  .2 

17 

2   0 

10  .8 

•i:]  .2 

37 

53 

30 

4  .9 

19  .G 

44  .1 

18 

2 

10  .9 

43  .6 

38 

54 

32 

5  .0 

20  .0 

45  .0 

20 

5 

11  .0 

44  .0 

39 

50 

35 

5  .1 

20  .4 

45  .9 

22 

7 

11  .1 

44  .4 

40 

58 

37 

5  .2 

20  .8 

46  .8 

23 

10 

11  .2 

44  .8 

41 

59 

40 

5  .3 

21  .2 

47  .7 

25 

12 

11  .3 

45  .-J 

42 

3   1 

42 

5  .4 

21  .G 

48  .G 

26 

15 

11  .4 

45  .6 

43 

2 

45 

5  .5 

22  .0 

49  .5 

28 

17 

11  .5 

46  .0 

44 

4 

47 

5  .0 

22  .4 

50  .4 

30 

20 

11  .6 

40  .4 

44 

6 

50 

5  .7 

22  .8 

51  .3 

31 

22 

11  .7 

46  .8 

45 

7 

52 

5  .8 

23  .2 

52  .2 

33 

25 

11  .8 

47  .2 

40 

y 

55 

5  .9 

23  .6 

53  .1 

34 

27 

11  .9 

47  .0 

47 

10 

5T 

6  .0 

24  .0 

54  .0 

36 

30 

12  .0 

48  .0 

48 

12 

_5  0  j 

TABLE   XVIII.— VOLUME   FOR  LENGTH   100  FEET. 
BASK  =  8.     SIDE  ST.OPES  2  TO  1. 


D 

0 

.1 

.2 

.:; 

4 

i  .  5 

<; 

.7  |  .S 

.9 

0 

{ 

I 

K 

u 

17 

20 

24 

2e 

33 

1 

3" 

4; 

4( 

51 

56 

61 

66 

72 

r»? 

83 

2 

8£ 

9E 

101 

107 

114 

120 

127 

154 

141 

148 

3 

156 

m 

171 

178 

186 

194 

203 

211 

220 

228 

4 

23? 

246 

255 

264 

274 

283 

293 

303 

313 

323 

5 

333 

344 

354 

365 

376 

387 

398 

410 

421 

433 

6 

414 

456 

468 

481 

493 

506 

518 

531 

544 

557 

570 

584   597 

611 

625 

639 

653 

667 

682 

696 

8 

711 

726 

741 

756 

WWfl 

787 

803 

818 

834 

850 

9 

8(i7 

883 

900 

916 

933 

950 

967 

984 

1002 

1019 

to 

1037 

1055 

1073 

1091 

1109 

1128 

1146 

1165 

1184 

1203 

11 

1222 

1242 

1261 

1281 

1300 

1320 

13401  1361 

1381 

1402 

12 

1422 

1443 

1  1(54 

1485 

1506 

1528 

1549 

1571 

1593 

1615 

18 

16:57 

16591  1(582 

1704 

1  727 

1750 

1773 

1796 

1820 

1843 

14 

1867 

1890)  1914 

1938 

1963 

1987 

2012 

2036 

2061 

2086 

15 

2111 

2136 

2162 

2187 

2213 

2239 

2265 

2291 

2317 

2344 

16 

2370 

2397 

2424 

2151 

2478 

2506 

2533 

2561 

2588 

2616 

17 

2614!  2673 

2701 

2730 

2758 

2787 

2816 

2845 

2874 

2904 

18 

2933  :  2963 

','993 

302: 

3053 

3083 

3114 

3144 

3175 

3206 

19 

3237 

3268 

3300 

3331 

3363 

3394 

3426 

3458 

3491 

3523 

20 

3556 

3588 

3621 

3654 

3687 

3720 

3754 

3787 

3821 

3855 

21 

3889 

3923 

3957 

3992 

4026 

4061 

4096  4131 

4166 

4202 

2-2 

4237 

4273 

4308 

4344 

4380 

4417 

4453   4490 

4526 

4563 

23 

4  GOO 

4637 

4(i74 

4712 

4749 

4787 

4^2;V  4863 

4901 

4939 

24 

4978  !  501(1 

5055 

5094 

5133 

5172 

5212 

5251 

5291 

5380 

OK 

53?0:  5410 

5451 

5491 

5532 

5572 

5613 

5654 

5695 

5736 

26    t 

5778 

5819 

5861 

5<)03 

5915 

5987 

6029   6072 

6114 

6157 

Off 

6200 

6243 

6286 

6330 

(5373 

6417 

6460 

6504 

6548 

6593 

28 

6637 

6682 

6726 

6771 

6816 

6861 

6906 

6952 

6997 

7043 

29 

7089 

7135 

7181 

7227 

7274 

7320 

7367 

7414 

7461 

7508 

BASE  =  8.  SIDE  SLOPES  3  TO  1. 

0 

3 

6 

10 

14 

18 

22 

26 

31 

36 

1 

41 

46 

52 

57 

63 

69 

76 

82 

89 

96 

2 

104 

111 

119 

127 

135 

144 

152 

161 

170 

179 

3 

1S9 

199   209 

219 

229 

240 

251 

262 

273 

285 

4 

296 

308   320 

333 

345 

358 

371 

385 

398 

412 

5 

426 

440   455 

469 

484 

499 

514 

530 

546 

562 

6 

578 

594 

611 

628 

645 

6()2 

680 

697 

715 

733 

752 

770 

789 

808 

828 

847 

867 

887 

907 

928 

8 

948 

969 

990 

1011 

1033 

1055 

1077 

1099 

1121 

1144 

0 

1167 

1190 

1213 

1237 

1260 

1284 

1308 

1333 

1357 

1382 

10 

1407 

1433 

1458 

1484 

1510 

1536 

1563 

1589 

1616 

1643 

1! 

1(570 

1698 

1726 

1754 

1782 

1810 

1839 

1868 

1897 

1926 

12 

1956 

1985 

2015 

2045 

2070 

2106 

2137 

2168 

2200 

2231 

13 

2263 

2295 

2327 

2360 

2392 

2425 

2458 

2491 

2525 

2559 

14 

2593 

2627 

2661 

2U96 

2731 

2766 

2801 

2837 

2872 

2908 

15 

2914 

2981 

3017 

3054 

3091 

3129 

3166   3204 

3242 

3280 

16 

3319 

3357 

3396 

3435 

3474 

3514 

3554 

3594 

3634 

3674 

17 

3715 

3756 

3797 

3838 

38SO 

3921 

3963 

4005 

4048 

4090 

18 

4133 

4176 

4220 

4263 

4307 

4351 

4395 

4440 

4484 

4529 

19 

4574 

4619 

4665 

4711 

4757 

4803 

4849 

4896 

4943 

4990 

20 

5087 

5085 

5132 

5180 

5228 

5277 

5325 

5374 

5423 

5473 

21 

5522 

5572 

5622 

5072 

5723 

5773 

5824 

5875 

5926 

5978 

22 

6030 

6082   6134 

6186 

6239 

6292 

6345 

6398 

6452 

6505 

23 

6559 

6613 

66C8 

6722 

6777 

6832 

6888 

(5943 

6999 

70"  5 

24 

7111 

7168 

7224 

7281 

7338 

7395 

7453 

7511 

7569 

7627 

25 

7085 

7744 

7803 

7862 

?(.'21 

7981 

8040 

HI  00 

8160 

8221 

26 

8281 

8342 

SI  03 

8485 

8f>2(; 

8588 

8650 

87'  12 

8775 

8837 

27 

8900 

8963 

9026 

U090 

9154 

9218 

9282 

9346 

9411 

9476 

28 

9r>41 

9C06 

96W 

MK 

HSO? 

9869 

9986 

10002 

10069 

10136 

29 

10204 

10271 

10339 

10407 

10475 

105441  106121  10681 

10750 

10819 

TABLE  xvni.— VOLUME  FOR  LENGTH  100  FEET. 

BASE  =14.     SIDE  SLOPES  f  TO  1. 


D 

.0 

.1 

.2    .3    .4 

.5 

.6    .7 

.s 

.9 

0 

5 

11 

16 

22 

27 

33 

39 

45 

51 

1 

57 

64 

70 

Mjff 

83 

90 

97 

104 

111 

119 

2 

126 

133 

141 

149 

156 

1G4 

172 

181 

IS!) 

197 

3 

206 

214 

223 

232 

241 

250 

259 

2(58 

277 

287 

4 

296 

306 

316 

826 

336 

346 

356 

306 

377 

887 

5 

398 

409 

420 

431 

442 

453 

4(55 

476 

488 

499 

6 

511 

523 

535 

547 

559 

572 

584 

597 

609 

G22 

y 

635 

648 

661 

675 

688 

701 

715 

729 

742 

75G 

8 

770 

785 

799 

813 

828 

842 

857 

887 

902 

9 

917 

932 

947 

903 

978 

994 

1010 

1096 

1042 

1058 

10 

1074 

1090 

1107 

1123 

1140 

1157 

1174 

1191 

1208 

1225 

11 

1243 

12GO 

1278 

1295 

1313 

1331 

1349 

1367 

1385 

1404 

19 

1422 

1441 

1459 

1478 

1497 

1536 

1535 

1555 

1574 

1593 

13 

1613 

1633 

1652 

1672 

1692 

1713 

1733 

1753 

1771 

1794 

14 

1815 

1836 

1857 

1878 

1  Si>9 

1920 

1941 

1963 

1984 

2006 

15 

2028 

2050 

2072 

2094 

2116 

2138 

2161 

2183 

2206 

8229 

16 

2252 

2275 

2298 

2321 

2345 

2368 

2392 

2415 

2439 

2468 

17 

2487 

2511 

2535 

2560 

2584 

2C09 

2633 

2(558 

2683 

2708 

18 

2733 

2759 

2784 

2809 

2835 

2861 

2886 

2912 

2938 

29G5 

19 

2991 

3017 

3044 

3070 

3097 

3124 

3151 

3178 

3205 

3232 

20 

3259 

3287 

3314 

3342 

3370 

3398 

3426 

3454 

3482 

3510 

21 

3589 

3567 

3596 

3625 

3654 

3683 

3712 

3741 

3771 

8800 

22 

8830 

3859 

38*9 

3919 

3949 

3979 

4009 

4040 

4070 

4101 

23 

4131 

4162 

4193 

4224 

4255 

4287 

4318 

4349 

4381 

4413 

24 

4444 

4476 

4508 

4541 

4573 

4(505 

4638 

4670 

4703 

4736 

25 

4769 

4802 

4835 

4868 

4901 

4935 

4968 

5002 

5036 

5070 

26 

Oft 

5104 
5450 

5138 

5485 

5172 
5521 

5206 
5556 

5241 
5592 

5275 

5627 

5310 
5663 

5345 
5699 

5380 
5735 

5415 
5771 

28 

5807 

5844 

5880 

5917 

5953 

5990 

6027 

6064 

6101 

6139 

29 

6176 

6213 

6251 

6289 

6326 

63G4 

6402 

6441 

6479 

6517 

BASE  =  10.  SIDE  SLOPES  1  TO  1. 

0 

4  !    8  '   11 

15 

19 

24 

28    32  I    3G 

1 

41 

45    50    54 

59 

64 

69 

74 

79     H4 

2 

89 

91    99   105 

110 

116 

121 

127 

133    139 

3 

144 

150   156 

163 

169 

175 

181 

188 

194  !   201 

4 

207 

214   221 

228 

235 

242 

249 

256 

263  ,   270 

5 

278 

285   293 

300 

308 

316 

324 

331 

339    347 

G 

356 

364 

372 

380 

389 

397 

406 

414 

423 

432 

7 

441 

450  !  459 

468 

477 

486 

495 

505 

514 

524 

8 

533 

543 

553 

563 

572 

582 

592 

603 

613 

623 

9 

633 

644 

654 

665 

675 

686 

697 

708 

719 

730 

10 

741 

752 

763 

774 

786 

797 

809 

820 

832 

844 

11 

856 

867 

879 

891 

904 

916 

928 

940 

953 

965 

12 

978 

990 

1003 

1016 

1029 

1042 

1055 

1068 

1081 

1094 

13 

1107 

1121 

1134 

1148 

1161 

1175 

1189 

1203 

1216 

1230 

14 

1244 

1259 

1273 

1287 

1301 

1316 

1330 

1345 

1359 

1374 

15 

1389 

1404 

1419 

1484 

1449 

1464 

1479 

1494 

1510 

1525 

10 

1541 

1556 

1572 

1588 

1604 

1619 

1635 

1651 

1668 

1684 

17 

1700 

1716 

1733 

1749 

1766 

1782 

1799 

181G 

1833 

1S50 

18 

1867 

1884 

1901 

1918 

1935 

1953 

1970 

1988 

2005 

2023 

19 

2041 

2059 

2076 

2094 

2112 

2131 

2149 

2107 

2185 

2204 

20 

22°2 

2241 

2259 

2278 

2297 

2316 

2335 

2354 

2373 

2302 

24U 

2430 

2450 

24ti9 

2489 

2508 

2528 

2548 

2568  I  2587 

22 

2607 

2027 

2648 

2668 

2688 

2708 

2729 

2749 

2770  !  2790 

23 

2811 

2832 

2853 

2874 

2895 

2916 

2937 

2958 

2979 

3001 

24 

3022 

3044 

3065 

3087 

3109 

3131 

3152 

3174 

3196 

3219 

25 

3241 

3263 

3285 

3308 

3330 

3353 

3375 

3398 

3421 

8444 

26 

3467 

3490 

3513 

3536 

3559 

3582 

3606 

3629 

3G53 

3676 

27 

3700 

3724   37  IS 

3771 

3795 

3819 

3844 

3868 

3892 

3916 

28 

3941 

3965 

3990 

4014 

4039 

4064 

4089 

4114 

4139 

4164 

29 

4189 

4214 

4239 

4265 

4290 

4316 

4341 

4367 

4393 

4419 

TAfcLE  XVlll.— VOLUME   FOR   LENGTH   100  FEET. 
BASE  =  16.    SIDE  SLOPES  f  TO  1. 


335 


D 

.0 

.1 

.2 

.3 

.4 

.6 

.6 

.  7 

.8 

.9 

0 

6 

12 

18 

25 

31 

38 

44 

51 

58 

1 

65 

r"*2 

79 

86 

94 

101 

109 

117 

125 

133 

2 

141 

149 

157 

166 

174 

183 

192 

201 

209 

219 

3 

228 

237 

247 

256 

266 

275 

285 

295 

305 

316 

4 

326 

336 

347 

358 

368 

379 

390 

401 

412 

424 

5 

435 

447 

458 

470 

482 

494 

506 

518 

531 

543 

556 

568 

581 

594 

607 

620 

633 

646 

660 

673 

ij* 

687 

701 

715 

729 

743 

757 

771 

786 

800 

815 

8 

830 

845 

859 

875 

890 

905 

921 

936 

952 

967 

9 

983 

999 

1015 

1032 

1048 

1064 

1081 

1098 

1114 

1131 

10 

1148 

1165 

1182 

1200 

1217 

1235 

1252 

1270 

1288 

1306 

11 

1324 

1342 

1361 

1379 

1398 

1416 

1435 

1454 

1473 

1492 

12 

1511 

1530 

1550 

1569 

1589 

1609 

1629 

1649 

1669 

1689 

13 

1709 

1730 

1750 

1771 

1792 

1813 

1833 

1855 

1876 

1897 

14 

1919 

1940 

1962 

1983 

2005 

2027 

2049 

2072 

2094 

2116 

15 

2139 

2162 

2184 

2207 

2230 

2253 

2276 

2300 

23-23 

2347 

16 

2370 

2394 

2418 

2442 

2466 

2490 

2515 

2539 

2564 

2588 

17 

2613 

2638 

2663 

2688 

2713 

2738 

2764 

2789 

2815 

2841 

18 

2867 

2893 

2919 

2945 

2971 

2998 

3024 

3051 

3078 

3105 

19 

3131 

3159 

3186 

3213 

3241 

3268 

3296 

3323 

3351 

3379 

20 

3407 

3436 

3464 

3492 

3521 

3550 

3578 

3607 

3636 

3665 

21 

3694 

3724 

3753 

3783 

3812 

3842 

3872 

3902 

JJ932 

3962 

22 

3993 

4023 

4054 

4084 

4115 

4146 

4177 

4208 

4239 

4270 

23 

4302 

4333 

4365 

4397 

4429 

4461 

4493 

4525 

4557 

4590 

24 

•1622 

4655 

4688 

4721 

4753 

4787 

4820 

4853 

4887 

4920 

25 

49:>4 

4987 

5021 

5055 

5089 

5124 

5158 

5192 

5227 

5262 

26   „ 

5296 

5331 

5366 

5401 

5436 

5472 

5507 

5543 

5578 

5614 

27 

5650 

5686 

5722 

5758 

5795 

5831 

5868 

5904 

5941 

5978 

28 

6015 

605-2 

6089 

6126 

6164 

6201 

6239 

6277 

6315 

6353 

29 

6391 

6429 

6467 

6506 

6544 

6583 

E622 

6661 

6699 

6739 

BASE  =  18.  SIDE  SLOPES  1  TO  1. 

0 

r- 

13 

20 

27 

34 

41 

48 

56 

63 

1 

70 

78 

85 

93 

101 

108 

116 

124 

132 

140 

2 

148 

156 

165 

173 

181 

190 

198 

207 

216 

224 

3 

233 

242 

251 

260 

269 

279 

288 

297 

307 

316 

4 

326 

336 

345 

355 

365 

375 

385 

395 

405 

416 

5 

426 

436 

447 

457 

468 

479 

489 

500 

511 

522 

6 

533 

544 

556 

567 

578 

590 

601 

613 

625 

636 

648 

660 

672 

684 

696 

708 

721 

733 

745 

758 

8 

770 

783 

796 

808 

821 

834 

847 

860 

873 

8S7 

9 

900 

913 

927 

940 

954 

968 

981 

995 

1009 

1023 

10 

1037 

1051 

1065 

1080 

1094 

1108 

1123 

1137 

1152 

1167 

11 

1181 

1196 

1211 

1226 

1241 

1256 

1272 

1287 

1302 

1318 

12 

1333 

1349 

1365 

1380 

1396 

1412 

1428 

1444 

1460 

1476 

13 

1493 

1509 

1525 

1542 

1558 

1575 

1592 

1608 

1625 

1642 

14 

1659 

1676 

1693 

1711 

1728 

1745 

1763 

1780 

1798 

1816 

15 

1833 

1851 

1869 

1887 

1905 

1923 

1941 

1960 

1978 

1996 

16 

2015 

2033 

2052 

2071 

2089 

2108 

2127 

2146 

2165 

2184 

17 

2204 

2-223 

2242 

2262 

2-281 

2301 

2321 

2340 

2360 

2380 

8 

2400 

24-20 

2440 

2460 

2481 

2501 

2521 

2542 

2562 

2583 

19 

2604 

2624 

2645 

2666 

2687 

2708 

2729 

2751 

2772 

2793 

20 

2815 

2836 

2858 

2880 

2901 

2923 

2945 

2967 

2989 

3011 

21 

3033 

3056 

3078 

3100 

3123 

3145 

3168 

3191 

3213 

3236 

22 

3259 

3282 

3305 

3328 

3352 

3375 

3398 

3422 

3445 

3469 

23 

3493 

3516 

3540 

3564 

3588 

3612 

3636 

3660 

3685 

3709 

24 

3733 

3758 

3782 

3807 

383-2 

3856 

3881 

3906 

3931 

3956 

25 

3981 

4007 

4032 

4057 

4083 

4108 

4134 

4160 

4185 

42'1 

26 

4237 

42(53 

4-2S9 

4315 

4341 

43f)8 

4394 

4420 

4447 

4473 

27 

4500 

-15-27 

4553 

4580 

4607 

4634 

4661 

4688 

471(5 

4743 

28 

4770 

4798 

48-25 

-1853 

4F8I 

4!K)8 

4S36 

4964 

4992 

5020 

29 

5048 

5076 

5105 

5133 

5161 

5190 

5218 

5247 

5276 

5304 

3B6    TABLE  xvin—  VOLUME  FOR  LENGTH  MO  FEET, 

BASE  =  20.    SIDE  SLOPES  1  TO  1. 


I) 

0 

t 

a 

;j 

4. 

t 

§ 

§ 

<j 

0 

7 

15 

2o 

30 

38 

46 

54 

62 

1 

78 

80 

94 

103 

111 

11) 

128 

137 

145 

1 

2 

103 

172 

181 

190 

199 

20S 

218 

gj» 

230 

2 

3 

866 

265 

275 

285 

295 

305 

315 

3-J5 

335 

3 

4 

356 

36(5 

376 

3sr 

398 

408 

419 

430 

411 

4 

5 

403 

474 

485 

497 

508 

519 

531 

543 

554 

5 

G 

578 

590 

602 

614 

626 

638 

650 

003 

675 

0 

7 

700 

713 

725 

738 

751 

764 

790 

803 

8 

8 

830 

8-13 

856 

870 

884 

897 

911 

9-J5 

939 

9 

9 

967 

981 

995 

1009 

1024 

1038 

1052 

1007 

1082 

10 

10 

1111 

1  126 

1141 

1156 

1171 

1186 

1201 

1217 

1232 

12 

11 

1263 

1279 

1294 

1310 

1326 

1342 

135S 

1374 

1390 

14 

12 

1422 

1439 

1455 

1471 

14S8 

1505 

1521 

1538 

1555 

15 

13 

1589 

1606 

1623 

1640 

1658 

1IJ75 

1692 

1710 

172S 

17 

14 

1703 

1781 

1799 

1817 

1835 

1853 

1871 

1889 

1908 

19 

15 

1944 

1963 

1982 

2000 

2019 

2038 

2057 

^076 

2095 

21 

10 

2133 

2153 

2172 

2191 

2211 

2231 

2250 

2270 

2290 

23 

17 

2330 

2350 

2370 

2390 

2410 

2431 

2451 

247! 

249;.' 

25 

18 

2533 

2654 

2575 

2590 

2617 

2638 

2059 

26-50 

2702 

27 

19 

2744 

2766 

2788 

2809 

2831 

2853 

2875 

2897 

2919 

29 

20 

2963 

2985 

3008 

3030 

3052 

3075 

3098 

3120 

3143 

31 

21 

3189 

32  W 

3235 

3258 

3281 

3305 

3328 

33;)! 

3375 

33 

22 

3422 

3446 

3470 

3494 

3518 

3542 

3566 

3590 

3614 

36 

23 

3003 

3687 

3712 

3737 

3761 

3786 

3811 

3836 

3861 

38 

24 

3911 

3936 

3962 

3987 

4012 

4038 

4064 

4089 

4115 

41 

25 

4167 

4193 

4219 

4245 

4271 

4297 

4324 

4350 

4376 

44 

26 

4430 

4456 

4483 

4510 

4537 

4564 

4591 

4618 

4W5 

46 

27 

4700 

4727 

4755 

4783 

4810 

4838 

4806 

4894 

49','2 

49. 

28 

497S 

5006 

5034 

5063 

5091 

5119 

5148 

5177 

5205 

52. 

29 

5263 

5292 

5321 

5350 

5379 

5408 

5438 

5467 

5497 

55 

BASE  =  22.    SIDE  SLOPES  1  TO  1. 


0 

8 

16 

25 

33 

42 

50 

59 

68 

7'6 

1 

85 

94 

103 

112 

121 

131 

140 

149 

159 

168 

jj 

178 

187 

197 

207 

217 

227 

237 

247 

257 

207 

3 

278 

288 

299 

309 

320 

331 

341 

352 

363 

374 

4 

385 

396 

408 

419 

430 

442 

453 

-105 

476 

488 

5 

500 

512 

524 

536 

548 

500 

572 

585 

597 

610 

6 

022 

035 

048 

060 

673 

686 

699 

712 

725 

739 

7 

752 

788 

779 

792 

806 

819 

833 

847 

801    875 

8 

889 

903 

917 

931 

916 

900 

975 

989 

1004 

1019 

9 

1033 

1048 

1063 

1078 

1093 

1108 

1124 

1139 

1154 

1170 

10 

1185 

1201 

1210 

1  '.'32 

1-M8 

1204 

1280 

1296 

1312 

1328 

11 

1344 

1361 

1377 

1  :;9  1 

1410 

1427 

1444 

1400 

1477 

1494 

12 

1511 

1528 

15-15 

1503 

1580 

15!  1  7 

1615 

1633 

1650 

1667 

13 

1685 

1703 

1721 

1739 

1757 

1  775 

1793 

1811 

1830 

1848 

14 

1867 

1885 

1904 

11)23 

1941 

1900 

1979 

1998 

2017 

2086 

15 

2056 

207'5 

2094 

2114 

2133 

2153 

2172 

2192 

2212 

2232 

16 

»>>>'•)•> 

2272 

2292 

2312 

2332 

2353 

2373 

23'.)4 

2414 

2435 

17 

2456 

2476 

2497 

251H 

2539 

2560 

2581 

2603 

2624 

2015 

18 

2667 

£688 

2710 

2731 

2753 

2775 

2707 

2819 

2841 

2863 

19 

2885 

2907 

2930 

2H52 

2975 

2997 

3020 

3043 

3065 

8088 

20 

3111 

3134 

3157 

3180 

3204 

3227 

3250 

3274 

:;-ji)7 

3321 

21 

3344 

3368 

3392 

3416 

3440 

3404 

3488 

3512 

3530 

3501 

22 

3585 

3610 

3634 

3659 

3684 

8708 

3733 

3758 

3783 

3808 

23 

3833 

3859 

3884 

3909 

3935 

3960 

3986 

4011 

4037 

4063 

24 

4089 

4115 

4141 

4107 

1193 

4219 

4240 

4-.'72 

4299 

4325 

25 

4352 

4319 

4405 

4132 

4459 

4488 

4513 

4540 

4568 

4595 

*26 

4622 

4650   1677 

4705 

4732 

4760 

4788 

4^16 

1844 

4872 

27 

4900 

4928 

4956 

4985 

5013 

5012 

5070 

5099 

5128 

5156 

28 

5185 

5214 

5243 

5272 

r.30i 

5331 

5360 

5389 

5419 

5448 

29 

5478 

5507 

5537 

5567 

5597 

5627 

5657 

5687 

5717 

5747 

TABLE  XIX.— VOLUME  FOR  LENGTH   100  FEET. 
BASE  =  0.     SIDE  SLOPES  1  TO  1. 


337 


D 

.0 

.1 

2 

.3 

.4 

.5 

.6 

.  7 

.8 

.9 

0 

1 

1 

1 

2 

0 

3 

1 

4 

4 

5 

6 

8 

9 

11 

12 

13 

8 

15 

16 

18 

20 

21 

23 

25 

27 

29 

31 

3 

33 

36 

38 

40 

43 

45 

48 

51 

53 

56 

4 

59 

62 

65 

68 

72 

75 

78 

82 

85 

89 

98 

96 

100 

104 

108 

112 

116 

120 

1  25 

129 

6 

133 

138 

142 

147 

152 

156 

161 

166 

171 

176 

181 

187 

198 

197 

203 

208 

214 

220 

2-25 

231 

8 

237 

243 

249 

255 

2(51 

268 

274 

280 

287 

SP8 

g 

800 

307 

313 

sao 

327 

334 

341 

348  |  350 

363 

10 

370 

378 

385 

393 

401 

408 

416 

424  1  432 

440 

11 

448 

456 

466 

473 

481 

490 

498   507  i  516 

524 

12 

533 

542 

551 

560 

509 

579 

588 

597  ;  C07 

616 

13 

6-26 

636 

645 

655 

665 

675 

685 

695 

705 

716 

14 

7-26 

736 

747 

757 

768 

779 

789 

800' 

811 

822 

15 

833 

S44 

856 

867 

878  1  890 

901 

913 

925 

936 

16 

948 

960 

972 

984 

996 

1008 

1021 

1033 

1045 

1058 

17 

1070 

1083 

1096 

1108 

1121 

1134 

1147 

1160 

1173 

1187 

18 

1200 

1213 

1227 

1:240 

1254 

1268 

1-281 

15-95 

1309 

1323 

19 

1337 

1351 

1305 

1380 

1394 

1408 

1423 

1437 

1452 

1467 

20 

1481 

1496 

1511 

1556 

1541 

1556 

1572 

1587 

1602 

1618 

21 

1633 

1649 

1665 

16SO 

1696 

1712 

1728 

1744 

1760 

1776 

22 

1793 

1809 

1825 

1842 

1858 

1875 

1892 

1908 

1925 

1942 

23 

1959 

1976 

1993 

2011 

20-28 

2045 

2063 

2080 

2098 

2116 

24 

2133 

2151 

2169 

2187 

2205 

2223 

2241 

2260 

2278 

'J296 

25 

2314 

2333 

2352 

2371 

2389 

2408 

2427 

244(J 

2465 

2484 

26 

2504 

2523 

2542 

25(52 

2581 

2601 

2021 

2640 

2060 

2680 

27 

2700 

2720 

2740 

2700 

2781 

2801 

2821 

2842 

2S63 

2883 

28 

2904 

2924 

2945 

2966 

2987 

3008 

3029 

3051 

3072 

3093 

29 

3115 

3136 

3158 

3180 

3201 

3223 

324  b 

3267 

3289 

3311 

BASE  =  0.  SIDE  SLOPES  f  TO  1. 

0 

1 

1 

1 

2 

3 

4 

5 

1 

6 

y 

8 

9 

11 

13 

14 

16 

18 

20 

a 

22 

25 

27 

29 

32 

35 

38 

41 

44 

47 

3 

50 

53 

57 

61 

64 

68 

72 

76 

80 

85 

4 

89 

93 

98 

103 

108 

113 

118 

123 

128 

133 

5 

139 

145 

150 

156 

162 

168 

174 

181 

187 

193 

6 

200 

207 

214 

221 

228 

235 

242 

249 

257 

265 

7 

272 

280 

288 

293 

304 

313 

321 

3:29 

338 

347 

8 

836 

365 

374 

383 

392 

401 

411 

421 

430 

440 

9 

450 

460 

470 

481 

491 

501 

512 

523 

534 

545 

10 

556 

567 

578 

589 

601 

613 

624 

636 

648 

6CO 

11 

072 

685 

697 

709 

722 

735 

748 

761 

774 

787 

12 

800 

813 

827 

841 

854 

868 

882 

896 

910 

925 

13 

939 

953 

968   983 

998 

1013 

1028 

1043 

1058 

1073 

14 

1089 

1105 

1120 

1136 

1152 

1168 

1184 

1201 

1217 

1233 

15 

1250 

1267 

1284 

1301 

1318 

1335 

1352 

1309 

1387 

1405 

16 

1422 

1440 

1458 

1476 

1494 

1513 

1531 

1549 

1508 

1587 

17 

1606 

1625 

1644 

1663 

1682 

1701 

1721 

1741 

1760 

1780 

18 

1800 

1820 

1840 

1861 

1881 

1901 

1922 

1943 

1964 

1985 

19 

2006 

2027 

2048 

2069 

2091 

2113 

2134 

2156 

2178 

2200 

20 

2222 

2245 

2267 

2289 

2312 

2335 

2358 

2381 

2404 

2427 

21 

2450 

2473 

2497 

2521 

2544 

2568 

2592 

2616 

2640 

2665 

22 

2689 

2713 

2738 

2763 

2788 

2813 

2838 

2863 

2888 

2913 

23 

2939 

2965 

2990 

3016 

3042 

3068 

3094 

3121 

3147 

3173 

24 

3200 

3227 

3254 

3281 

3308 

3335 

33fc2 

3389 

3417 

3445 

25 

3472 

3500 

3528 

3556 

3584 

3613 

3641 

3069 

3098 

3727 

26 

3756 

3785 

3814 

3843 

3872 

3901 

3931 

3961 

3990 

4020 

27 

4050 

4080 

4110 

4141 

4171 

4201 

4232 

4263 

4x'94 

4325 

28 

4356 

4387 

4418 

4449 

4481 

4513 

4544 

4576 

4608 

4640 

29 

4672 

4705 

4737 

4769 

4802 

4835 

4868 

4901 

4934 

4967 

TABLE  XX. -SINES  AND  COSINES. 


0° 

1° 

2° 

30             40 

Sine  Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine  |  Cosin 

\ 

0 

.00000 

One. 

.01745 

.99985 

".03490 

99939 

.05234 

91(863  .06976.99756:60 

1 

;  1-0029 

One. 

.01774 

.99984 

.03519 

.99938 

.05263 

99801  .070051.99754159 

2 

.00058 

One. 

.01803 

.99984 

.03548 

.99937 

.0-3292 

99860 

.07034  .99752  58 

3 

.00087 

One. 

.01832 

.99983 

.03577 

.99936 

.05321 

.99858; 

.070631.99750 

57 

4 

.00116 

One. 

.01862 

.99983 

.03606 

.99935 

.05350 

.99857  .07092  .99748 

56 

5 

.00145 

One. 

.01891 

.99982 

.03635 

.99934 

.05379 

.99855; 

.07121 

.99746 

55 

6 

.00175 

One. 

.01920 

.99982 

.03664 

.99933 

.05408 

.99854 

.07150 

.99744 

54 

7 

.00204 

One. 

.01949 

.99981 

.03093 

.99932 

.05437 

.99852 

.07179 

.99742 

53 

8 

.00243 

One. 

.01978 

.99980 

.03723 

.99931 

.05466 

.99851 

.07208 

.99740 

52 

9 

.00262 

One. 

.02007 

.99980 

.03752 

.99930 

.05495 

.99849 

.07237 

.99738 

51 

iO 

.00291 

One. 

.02036 

.99979 

.03781 

.99929 

.05524 

.99847 

.07266 

.99736 

50 

11 

.00320 

.99999 

.02065 

.99979 

.03810 

.99927 

.05553 

.99846 

.07295 

.99734 

49 

12 

.00349 

.99999! 

.02094 

.99978 

.03839 

.99926 

.05582 

.99844 

.07324 

.99731 

48 

13 

.00378 

.99999; 

.02123  .99977 

1  .03868 

.99925 

.05611 

.99842 

.07353 

.99729 

47 

14 

.00407 

.99999' 

.02152|.  99977 

1.03897 

.99924 

.05640 

.99841 

.07382 

.99727 

46 

15 

.00436 

.99999 

..021811.99976 

.03926 

.99923 

.05669 

.99839; 

.07411 

.99725 

45 

10 

.00465  '.99999 

.02211 

.99976 

.03955 

.99922 

.05698 

.99838: 

.07440 

.99723 

44 

17 

.00495 

.99999 

.02240 

.99975 

.03984 

.99921 

.05727 

.99836; 

.07469 

.99721 

43 

18 

.00524 

.99999 

.02269 

.99974 

.04013 

.99919 

.05756 

.99834 

.07498 

.99719 

42 

19 

.00553 

.99998 

.02298 

.99974 

.04042 

.99918 

.05785 

.99833| 

.07527 

.99716 

41 

20 

.00582 

.99998 

.02327 

.99973 

.04071 

.99917 

.05814 

.99831 

.07556 

.99714  40 

21 

.00611 

.99998 

.02356 

.99972 

.04100 

.99916 

.05844 

.99829 

.07585 

.99712 

39 

22 

.00640 

.99998 

.02385 

.99972 

.04129 

.99915 

.05873 

.99827! 

.07614 

.99710  as 

23 

.00669 

.99998 

.02414 

.99971 

.04159 

.99913 

.05902 

.99826  .(7643 

.99708  37 

24  .006981.1)9998 

.02443 

.99970 

.04188 

.99912 

.05931 

.99824  .07672 

.99705 

36 

25  .00727 

.99997 

.02472 

.99969 

.04217 

.99911 

.05960 

.99822  .07701 

.99703  35 

26 

.00756 

.99997 

.02501 

.99969 

.04246 

.99910 

.05989 

.99821  .07730 

.99701  |  34 

27 

.00785 

.99997 

.02530 

.99968 

.04275 

.99909 

.06018 

.99819  .07759 

.99699  33 

28 

.00814 

.99997 

.02560  .99967 

.04304 

.99907 

.06047 

.99817  .07788 

.99696 

32 

29 

.00844 

.99996 

.025891.99966 

.04333 

.99906 

.06076 

.99815  .07817 

.99694 

31 

30 

.00873 

.99996 

.02618 

.99966 

.04362 

.99905 

.06105 

.99813 

.07846 

.99692 

30 

31 

.00902 

.99996 

.02647 

.99965 

.04391 

.99904 

.06134 

.9981  2  ' 

.07875 

.99689 

29 

32 

.00931 

.99996 

.02676 

.99964 

.04420 

.99902 

.06163 

.99810 

.07904 

.99687 

28 

33 

.00960 

.99995 

.027051.99963 

.04449 

.99901 

.06192 

.99808  .07933 

.99685 

27 

34 

.00989 

.99995 

.02734 

.99963 

.04478 

.99900 

.06221 

.99806  .07962 

.99683 

26 

35 

.01018 

.99995 

.02763 

.99962 

.04507 

.99898 

.06250 

.99804 

.07991 

.99680 

25 

36 

.01047 

.99995 

.02792 

.99961 

!  .04536 

.99897 

.06279 

.99803 

.08020 

.99678  2-1 

37 

.01076 

.99994 

.02821 

.99960 

.04565 

.99896 

.06308 

.99801 

.08049 

.  99676  j  23 

38 

.01105 

.99994 

.02850 

.99959 

.04594 

.99894 

.06337 

.99799 

.08078 

.99673!  22 

39 

.01134 

.99994 

.02879 

.99959 

.04623 

.99893 

.06366 

.99797 

.08107 

.99671  i  21 

40 

.011G4 

.99993 

.02908 

.99958 

.04653 

.99892 

.06395 

.99795 

.08136 

.99668  20 

41 

.01193 

.99993 

.02938 

.99957 

.04682 

.99890 

.06424 

.99793 

.08165 

.99666 

19 

42 

.01222 

.99993 

.02967 

.99956 

.04711 

.99889 

.06453 

.99792 

.08194 

.99664 

18 

43 

.01251 

.99992 

.02996 

.99955 

.04740 

.99888 

.06482 

.99790 

.08223 

.99661 

17 

44 

.01280 

.99992 

.03025 

.99954 

.04769 

.99886 

.06511 

.99788 

.08252 

.99659 

16 

45 

.01309 

.99991 

.0305* 

.99953 

.04798 

.99885 

.06540 

.99786 

.08281 

.99857 

15 

46 

.01338 

.99991 

.03083 

.99952 

.04827 

.99883 

.06569 

.99781 

.08310 

.99654 

14 

47 

.01367 

.99991 

.03112 

.99952 

.04856 

.99882 

.06598 

.99782 

.08339 

.99652 

13 

48 

.01396 

.99990 

.03141 

.99951 

.04885 

.99881 

.06627 

.99780 

.08368 

.99649 

12 

49 

.01425 

.99990, 

.03170 

.99950  .  04914  .99879 

.06656 

.99778 

.08397 

.99647 

11 

50 

.01454 

.99989 

.03199 

.99949  .04943  .99878 

.06685 

.99776  : 

.08426 

.99644 

10 

51 

.01483 

.99989 

.03228 

.99948  1.04972 

.99876 

.06714 

.99774 

.08455 

.99642 

9 

52 

.01513 

.99989' 

.03-.T.7 

.99947 

!  .05001 

.99875 

.067431.99772 

.08484 

.99639 

8 

53 

-01542 

.99988 

.03286 

.99946  I.  05030 

.99873  .06773 

.99770 

.08513 

.99637 

7 

54 

.01571 

.99988 

.03316 

.99945  .05059 

.99872 

.06802 

.99768 

.08542 

.99635 

6 

55 

.01600 

.99987 

.0-3345 

.99944  1  .05088 

.99870 

.06831 

.99766 

.08571 

.99632 

5 

56 

.01629 

.99987 

.03374 

.99943  .05117 

.99869 

.06860 

.99764 

.08600 

.99630  4 

57 

.01658 

.99986 

.03403 

.99942  1.05146 

.99867 

.06889  .99762 

.08629 

.99627  8 

58 

.01687 

.99986; 

.03432 

-.99941  .05175 

.99866 

.06918  .99760 

.08658 

.99625  2  i 

59 

.01716 

.99985 

.03461 

.99940 

.05205 

.99864 

.06947  .99758 

.08687 

.99622  1 

60 

.01745 

.  99985  1 

.03490 

.99939  .05234 

.99863 

.06976!.  99756 

.08716  .99619 

0 

/ 

Cosin 

Sine  I 

Cosin 

Sine  i  Cosin 

Bine 

Cosin  Sine 

Cosin  Sine 

/ 

89° 

88'   H   87°      86» 

85» 

TABLE   XX. -SINES  AND   COSINES. 


I    5° 

6° 

7°    |!    8* 

9° 

Sine 

Cosin 

Sine  Cosin 

Sine  !  Cosin 

Sine 

Cosin 

Sine  |  Cosin 

g 

Tl  .0871(5 

.99619 

.10453  .99452 

.12187 

.99255 

.13917  .99027 

.15648 

.98769 

(50 

1  .08745 

.99617 

.10482  .99449 

.12216 

.99251  .13946  .99023 

.15672 

.98764 

59 

2 

.08774 

.99614 

.10511  .99446 

.12245 

.99248 

.13975L99019 

.15701 

.98760 

58 

3 

.08803 

.99612 

.10540  .99443 

.12274 

.99244 

.14004;.  99015 

.15730 

.98755 

57 

4 

.08831 

.99609 

.10569  .99440 

.18808 

.99240 

.14033  .99011 

i  .15758 

.98751 

r>6 

5 

.08860 

.99607 

.10597 

.99437 

.12331 

.99237 

.14061  .99006 

.15787 

.98740 

55 

6 

.08889 

.99604 

.10626 

.99434 

.12360 

.99233 

.14090  .99002 

.15816 

.98741 

54 

7 

.08918 

99602 

.10655 

.99431 

.12:389  .99230 

.14119 

.98998 

.15845 

.98737 

53 

8 

.08947 

.99599 

.10684 

.99428 

.12418  .99226 

.  14148  98994 

15873 

98732 

52 

9 

.08976  .99596 

.10713  .99424 

.12447  .99222 

.14177 

.98990 

.15902 

.98728 

51 

10 

.09005 

.99594 

.10742  .  99421  j 

.12475  .99219 

I 

.14205 

.98906 

.15931 

.98723 

50 

11 

.09034 

.99591 

.10771 

.99418 

.12504 

.99215 

.14234 

.98982 

i.  15959 

.98718 

40 

12 

.09063 

.99588 

.10800 

.99415 

.135331.99211 

.5)8978 

.15988 

.98714 

48 

13 

.09092 

.99586 

.10829 

.99412 

.12562!.  99208 

!  14292  .98973 

.16017 

.9870S 

47 

14 

.09121 

.99583 

.10858 

.99409 

.  12591  |.  99204 

.14320  .98969 

.16046 

.98704 

46 

15 

.09150 

.99580 

1.10887 

.99406 

.12620  |.  99200 

.14346 

.98965 

.16074 

.98700 

45 

16 

.09179  .9957'8 

.10916 

.99402 

.12649  .99197 

.14378 

.98961 

.16103 

.98695 

44 

17 

.09208 

.99575 

.10945 

.99399 

.12678  .99193 

.14407 

.98957 

.16132 

.98690 

43 

18  !  .09237 

.99572 

.10973 

.99396 

.127061.99189 

.14436 

.98953 

.16160 

.98686 

48 

19  .09^66 

.99570 

.11002 

.99:39.3 

.12735 

.99186 

.14464 

.98948  .16189 

.98681 

41 

20 

.09295 

.99567 

.11031 

.99390, 

.12764 

.99182 

.14493 

.98944:  .16218 

.98676 

40 

21 

.09324 

.99564 

.11060 

.99386 

.12793 

.99178 

.14522 

.98940  .16246 

.98671 

30 

22  .09353 

.99562 

.110891.99383 

.12822 

.99175 

.14551 

.98936  .16275 

.98667 

38 

23  .09382 

.99559 

.11118  .99380 

.12851 

.99171 

.14580  .98931!  .16304 

.98662 

37 

24  .09411 

.99556 

.11147  .99377 

.12880 

.99167 

.  14608  i.  98927  1  .16333 

.98657' 

36 

25  .09440 

.99553 

.11176 

.99374 

.12908 

.99163 

.14637  >.98923i  .16361 

.98652 

39 

26 

.09469 

.99551 

.11205 

.99370 

.12937 

.99160 

.14666  .  98919  '  .16390 

.98648 

84 

27 

.09498 

.99548 

.11234 

.99367 

.12966 

.99156 

.14695!.  98914  .16419 

.98643 

38 

28 

.09527 

.99545 

.11263 

.99364 

.12995 

.99152 

.14723  .  98910  j  .16447 

.98638 

38 

29 

.09556 

.99542 

.11291 

.99360 

.13024 

.99148; 

.147521.98906  .16476 

.98633 

31 

30 

.09585 

.99540 

.11320 

.99357; 

.13053 

.  99144  j 

.147811.98902  .16505 

.98629 

.'50 

31 

.09614 

.99537 

.11349 

.  99354  ! 

.13081 

.99141 

.14810  .98897  i  .16533 

.98624 

29 

32 

.09642 

.99534 

.11378 

.99351 

.13110 

.99137 

.14838 

.98893  :  .16562 

.98619 

28 

33 

.09671 

.99531 

.11407 

.99:347 

.13139 

.99133 

.14867  .98889 

:  .16591 

.98614 

27 

34 

.09700 

.99528 

.11436 

.99344 

.13168 

.99129 

.148961.98884 

i  .16620 

.98609 

26 

35 

.09729 

.99526 

.11465 

.99341 

.13197 

.99125 

.14925 

.98880 

.16648 

.98604 

25 

36 

.09758 

.99523 

.11494 

.99337 

.13226 

.99122 

.14954 

.98876 

.16677 

.98600 

24 

37 

.09787 

.99520 

.11523 

.99334 

.13254 

.99118 

.14982 

.98871 

!.16706[.98595  23 

38 

.09816 

.99517 

.11552 

.99331 

.13283 

.99114 

.15011 

.98867 

.16734 

.98590 

22 

39 

.09845 

.99514 

.11580 

.99327 

.13312 

.99110 

.15040  .98863 

.16763 

.98585 

21 

40 

.09874 

.99511 

.11609 

.99324 

.13341 

.99106, 

.15069  .98858 

.16792 

.98580 

20 

41 

.09903 

.99508 

.11638 

.99320 

.13370 

.99102: 

.15097 

.98854 

.16820 

.98575 

19 

42  1  .09932 

.99506 

.11667 

.99317 

.13399 

.99098  .15126 

.98849 

.16849 

.98570 

18 

43  !  .09961 

.99503 

.11696 

.99314; 

.13427 

.99094 

.15155 

.98845 

.16878 

.98565 

17 

44  !  .09990 

.99500 

.11725 

.99310 

.13456 

.99091 

.15184  .98841 

.16906 

.98561 

16 

45 

.10019 

.99497 

.11754 

•99307 

.13485 

.99087! 

.15212;.  98836 

!  .16935 

.98556 

15 

46 

.10048 

.99494 

.11783 

.99303 

.13514 

.99083 

.15241  i  .98832 

i  .16964 

.98551 

14 

47 

.10077 

.99491 

.11812 

.99300 

.13543 

.99079 

.152701.98827 

.16992 

.98546 

18 

48 

.10106 

.99488 

.11840 

.99297 

.18572 

.99075 

.152991.98823 

.17021 

.98541]  12 

49 
50 

.10135 
.10164 

.99485 
.99482 

.11869 
.11898 

.99293 
.99290i 

.13600 
.13629 

.99071 
.99067 

.15327 
.15356 

.98818 
.98814 

.17050  .98536 
.17078  .98531 

11 
10 

51 

.10192 

.99479 

.11927 

.99286* 

.13658  '.99063 

.15385 

.98809 

.17107 

.98526 

9 

52 

.10221 

.99476 

1.11956 

.992831 

.13687  .99059 

.15414  .98805 

.17136 

.98521 

8 

53 

.10250 

.99473 

!ll985 

.99279 

.13716 

.99055 

.  15442  j.  98800 

.17164 

.98516 

7 

54 

.10279 

.99470 

.12014 

.99276 

.13744 

.99051 

.154711.98796 

.17193 

.98511 

6 

55 

.10308 

.99467 

.12043 

.99272 

.13773  .99047 

.15500  .98791 

.17222 

.98506 

5 

56 

.10337 

.99464 

.12071J.99269 

.138021.99043 

.155291.98787 

.17250 

.98501 

4 

57 

10366 

.99461 

!  .12100  .99265 

.13831  '.99039 

.15557  .98782  .17279 

.98496 

3 

58 

.10395 

.99458 

i.  12129  1.99262 

.138601.99035 

M55861.  98778  1.17308 

.98491 

2 

59 

.10424 

.99455 

!  .  12158  1  .  99258  .  13889  i  .  99031  : 

.15615  .98773 

.17336 

.98486 

1 

6(' 

.10453 

.99452 

!  .12187  .99255 

.13917  .99027 

.15643  .98769 

.17365 

.98481 

0 

t 

Sine 

Cosin  1  Sine  Cosin  \  Sine 

Cosin  Sine 

Cosin 

Sine 

r 

84* 

83° 

82'       81° 

80° 

TABLE  XX.— SINES  AND  COSINES, 


4 

10° 

11°       12° 

13° 

I    14° 

Sine  Cosin 

Sine  Cosin  !  Sine  Cosin 

Sine  Cosin 

Sine 

Cosin 

; 

"o 

717365  .'98481 

Tl908l 

.98163  1.20791  .97815 

.22495 

'!  97437 

.24198 

797030  i  fiO 

1 

.17393  .98476  .1910* 

.98157 

.20820  .97801) 

.22523  .974:30 

.24220 

.1)7023 

59 

2 

.17422  .98471 

.19138 

.98152 

.20848  .97803 

.22552 

.97424 

.24241) 

.97015 

,  58 

3 

.17451  .98466 

.19167 

.98146 

.20877 

.97797 

.22580  .97417 

.24277 

.97008 

;  57 

4 

.17479  .98461 

.  19195 

.98140 

.20905 

.97791 

.22608  .97411 

i  .24305 

.97001 

56 

5 

.17508  .98455 

.19224 

.98135 

.209:331.97784 

.22637 

.97404 

!  .24:333 

.96994 

55 

6 

.17537  .98450 

.19252 

.98129 

.20962  .97778 

.22665 

4)735  IS 

.24362 

.96987 

54 

7 

.17565  .98445 

.19281 

.98124 

.20990 

.97772 

.22693 

.97391 

.24390 

.96980 

53 

8 

.17591  .98440 

.19:309 

.98118 

.21019 

.97766 

.22722 

.973S4 

.24418 

.96973 

52 

9 

.17623  .98435 

.19338 

.98112 

.21047 

.97760 

.22750 

.97378 

.24446 

.96966 

51 

10  .17651;.  98430 

.19366 

.98107 

.21076 

.97754 

.22778 

.97371 

.24474 

.96959 

50 

11 

.17680 

.98425 

.19395 

.98101 

.21104 

.97748 

.22807 

.97365 

.24503 

.96952 

49 

12 

.17708 

.98120 

.19423 

.98096 

.21132 

.97742 

.22835 

.97358 

.24531 

.96945 

48 

13  .17737 

.98414 

.19452 

.98090 

.21161 

.97735 

.2286:3 

.97:351 

.24559 

.1)61)37 

47 

14  .17766 

.98409 

.19481 

.98084 

.21189 

.97729 

.22892 

.97345 

.24587 

961)30 

46 

15  .17794 

.98404 

.19509 

.98079 

.21218 

.97723 

.22920 

.97338 

.24615 

.96923 

45 

16  |.  17823 

.98399 

.19538 

.98073 

.21246 

.97717 

.22948 

.97331 

.24644 

.96916 

44 

17  .17852 

.98394 

.19566 

.98067 

.21275 

.97711 

.22977 

.97325 

.24672 

.96909 

43 

18 

.17880 

.983891 

.19595 

.98061J 

.21303 

.97705 

.23005 

.97318 

.24700 

.%<)(>•> 

42 

19 

.17909 

.98:383 

.19623 

93056 

.21331 

.97698 

.23033 

.97311 

.347281.96894 

41 

20 

.17937 

.98378 

.19652 

98050 

.21360 

.97692 

.23062 

.97'304 

.24750 

.96887 

40 

21 

.17966 

.98373 

.19680 

98044 

.21388 

.97686 

.23090 

.97298 

.24784 

.96880 

39 

22  1.1791)5  .98368 
23  .180231.98362 

.19709 
.19737 

98039 
93033 

.21417 
.21445 

.97680 
.97673 

.23118 
.23146 

.97291 

.97284 

;  .24813 
i  .24841 

.96873 
.1)6866 

38 
37 

24 

.13052 

.98:357 

.19766 

93027 

.21474 

.97667 

.23175 

.97278 

.24869 

.1)6858 

36 

25 

.18081 

.98352 

.19794 

98021 

.21502 

.97661 

.23203 

.97271 

.24897 

.96851 

35 

26 

.18109 

.98347 

.19823 

98016 

.21530 

.97655 

.23231 

.97'264 

.24925 

.96844 

34 

27 

.18138 

.98341 

.19851 

93010  '  .21559 

.97648 

.2,3260 

.97257! 

.24954 

.1)6837 

33 

28 

.18166 

.98a36 

.19880 

98004  !  .21587 

.97642 

.23288 

.97251! 

.24982 

.96829 

32 

29 

.18195 

.98331 

.19908 

97913  i  .21616 

.97636 

.23310  .97244; 

.25010 

.96822 

31 

30 

.18224 

.98325 

.19937 

97932  .21644 

.97630 

.23345  .97237 

.25038 

.96815 

30 

31 

.18252 

.98320 

.19965 

97987 

.21672 

97623 

.23373 

.97230! 

.25066 

.96807 

29 

32 

.18281 

.98315  .19994 

97981  .21701 

97617 

.2-3401 

.97223 

.25094 

.96800 

28 

33 

.18309 

.98310  .20022 

97975  !  .21729 

97311 

.23429 

.97217 

.25122 

.96793 

27 

34 

.18338 

.98304 

.20051 

97969  i  .21758 

97604 

.23458 

.97210! 

.25151 

.96786 

26 

35  .18367 

.98299  .20079 

97963  |  .21786 

97598 

.23486 

.97203 

.25179 

.96778 

25 

36  .18395 

.  93294  | 

.20108 

97958 

.21814 

97592 

.23514 

.97196 

.25207 

.96771 

24 

37 

.18424 

.98288 

.20136 

97952 

.21843 

97585 

.23542 

.97189 

.25235 

.96764 

23 

38 

.184521.98283 

.20165 

97946 

.21871 

97570 

.23571 

.97182 

.25263 

.96756 

22 

39 

.18481 

.  98277  ii  .20193 

97940 

.21839 

97573 

.23599 

.97176 

.25291  .96749 

21 

40 

.18509 

.98272!  .20222 

97934 

.21928 

97566, 

.23627 

.97169 

.25320  .96742 

20 

41 

.18538 

.98267 

.20250 

97928 

.21956 

97560 

.23656 

.97162 

.25348  .96734 

19 

42 

.18567;.98261 

.20279 

97922 

.21985 

97553 

.23684 

.97155 

.25376  .96727 

18 

43 

.18595|.98256 

.20307 

97916 

.22013 

97547 

.23712 

.97148 

.25404!.  9671  9 

17 

44 

.18624;.98250 

.20336 

97910! 

.22041 

97541 

.23740 

.97141' 

.25432  .96712 

16 

45 

.18652  .98245 

.20364 

97905  .22070 

97534 

.23769 

.97134 

.  25460  i.  96705 

15 

46 

.18681 

.98240 

.20393 

97899  .22098 

97528 

.23797 

.97127; 

.254881.96697 

14 

47 

.18710 

.98234 

.20421 

97893 

.22126 

97521 

.23825 

.97120 

.25516  .96690 

13 

48 

.18738 

.98229 

.20450 

97887 

.22155 

.97515 

.23853 

.97113 

.25545  .96682 

12 

49 

.18767 

.98223 

.20478 

97881 

.22183 

.97508 

.23882 

.97106 

.255731.96675 

11 

50 

.18795 

.98218  !  .20507 

.97875 

.22212  .97502 

.23910 

.  97100  j 

.25601  .96667 

10 

51 

.18824 

.98212 

.20535 

.978691 

.22240  .97496 

.23938 

.97093 

.25629  .96660 

9 

52 

.18852 

.98207 

.20563 

.97863 

.222681.97489 

.23966 

.97086 

.25657  .96653 

8 

53 

.18881 

.98201 

.20592  .97857 

.222971.97483  .23995 

.97079 

.25685  .96645 

7 

54 

.18910 

.98196  .20620  .97851 

.223251.97476  .24023 

.97072 

.25713  .96638 

6 

55 

.18938 

.98190  .20649 

.97845 

.223531.97470 

.24051 

.97065 

.25741  .96630 

5 

56 

.  18967  j.  98185  1  1.20677 

.97839 

.22382i.97463 

.24079 

.97058 

.25769  .96623 

4 

57 

.18995  .98179;  1.20706  .97833 

.224101.97457 

.24108 

.97051 

.25798  .96615 

3 

58 

.19024  .981741 

.20734 

.22438  .97450 

.241361.97044 

.25826  .96608 

2 

59 

.  19052  ;.  98168  j 

.20763  .97821 

.22467  .97444 

.241641.97037 

.25854  .96600 

1 

60 

.19081;.  98163 

.20791  .97815 

.22495  .97437. 

.24192  .97030 

.25882  .96593 

0 

Cosin  Sine  i  Cosin 

Sine 

Cosin 

Sine 

Cosin  |  Sine  | 

Cosin  Sine 

79°   II   78° 

77° 

76°   ! 

75° 

TABLE   XX.-S1NES    AND   COSINES. 


34 


15°       16°       17°   ; 

18° 

19° 

Sine  Cosin  i  Sine 

Cosin  Sine 

Cosin 

Sine 

Cosin 

I  Sine 

Cosin 

Ol.  25883  .90593  .275(54 

.96126  .29237 

.95630 

.30902 

.95106 

.32557 

.94552 

BC 

1  .25910  .96585 

.27592 

.90118  .29205 

.95022 

.30929 

.95CJ7 

.32584 

.94542 

59 

2  j.  25938  .96578 

.27020 

.90110  .29293  .95013 

.30957 

.95088 

'  .32612 

.94533;  58 

3  .25906:.  96570  : 

.27648 

.90102  .29321  .95605 

.30985 

.95079 

.32639 

.94523  57 

4  .25994  .96562 

.27676 

.90094  .29348  .95596 

.31012  .95070 

.32667 

.94514  58 

5  .26022;.  96555 

.27704 

.96086  .29376  .95588 

.31040  .95061 

.32694 

.94504  55 

6  .20050  .96547 

.27731 

.96078  .29404  .95579 

.310081.95052 

.32722 

.94495 

51 

7  .26079  .96540 

.27759 

•90070  .29432 

.95571 

.31095  .95043 

.32749 

.94485 

K 

8  .26107  .96532: 

.27787 

.96002  .29460 

.95502 

.  31123  1.95033 

.32777 

:  94470 

52 

9  '.26135  .96524 

.27815 

.9605-4'  .29487 

.95554' 

.31151 

.95024 

•  .32804 

.94460 

51 

10  .26163 

.  96517  j 

.27843 

.90040^.29515 

.95545 

.31178 

.95015 

.32832 

.94457 

•H 

11 

.26191 

.96509 

.27871 

.96037  .29543 

.95536! 

.31206  .95006 

'  .32859 

.91447 

49 

12 

.26219 

.96502 

.27899 

.96029 

.29571 

.95528 

.31233  .94997 

1  .32887 

.5X438 

4:^ 

13 

.26247  .96494 

.27927 

.96021 

.29599 

.9551  9  S 

.31201  .94988 

i  .32914 

.94428 

47 

14 

.26275  .96486 

.27955 

.96013;  .29626 

.95511! 

.31289  .94979 

.32942  .94418 

46 

15 

.26303  .96479 

.27983 

.96005  .29654  .955021 

.31316  .94970 

.32969  .94409 

« 

16 

.  26331  !.  96471 

.28011 

.95997 

.29682  .  95493  ( 

.31344  .94961 

.32997 

.94399 

44 

17 

.26359  .96463 

.28039 

.95989 

.29710  .95485! 

.31372  .94952 

.33024 

.94390 

43 

18 

.26387 

.96456 

.28067 

.95981 

.29737 

.95476 

.31399 

.94943 

i  .33051 

.94380 

42 

19 

.26415 

.96448 

.28095 

.95972 

.29765 

.95467! 

.31427  .94933 

1  .33079 

.94370 

41 

20 

.26443 

.96440 

.28123 

.95964 

.29793 

.95459; 

.31454 

.94924 

'.33106 

.94361 

K 

21 

.26471 

.  96433  ' 

.28150 

.95956 

.29821 

.95450 

.31482 

.94915 

.33134 

.94351 

3! 

22 

.26500 

.96425 

.28178 

.95948 

.29849 

.95441! 

.31510 

.94900 

i  .33161 

.94342 

88 

23  .26528 

.  96417  > 

.28206 

.95940 

.29876 

.95433 

.31537 

.94897 

.33189 

.94332  37 

24  1  .26556 

.96410 

.28234 

.95931 

.29904 

.95424 

.31565 

.94888 

.33216 

.94:WC  30 

25  '.26584 

.90402 

.28262 

.95923 

.29932 

.95415 

.31593 

.94878 

.33244 

.94313  35 

26  [  .26612 

.96394 

.28290 

.95915 

.29960 

.95407 

.31620 

.94869 

.33271 

.943031  34 

27  ;  .26640  .96386 

.28318 

.95907 

.29987 

.95398 

.31648 

.94860 

.33298  .94293  33 

28 

.26668  .96379 

.28340 

.95898 

.30015 

.95389 

.31675 

.94851 

.33326 

.<>42Ht  32 

29 

.26696  .96371 

.28374 

.95890 

[80043 

.95380 

.31703 

.94842 

.33353 

.94274  31 

30 

.26724  .96363 

.28402 

.95882 

.30071  .95372! 

.31730 

.94832 

.33381 

.94264 

30 

31 

.26752  .96355 

.28429 

.95874 

.30098  .95363 

.31758 

.94823 

.33408 

.94254 

21 

32 

.20780  .90347 

.28457 

.95865 

.30126  .95354 

.31786 

.94814 

.33436 

.94245 

28 

83 

.26808  .96340 

.28485 

.95857 

.30154  .95345 

.31813 

.94805 

:.  33463 

.94235 

27 

34 

.26836 

.96332 

.28513 

.95849 

.30182  .95337 

.31841 

.94795 

.33490 

.94225 

•>< 

35 

.26864  .96324 

.28541 

.95841 

.30209  .95328J 

.31868 

.94780 

i.  33518 

.94215 

•X 

36 

.26892  .96316 

.28509 

.95832 

.30237  .95319 

.31896 

.94777 

1  .33545 

.94200 

24 

37 

.26920 

.90308 

.28597 

.95824 

.30205  .953101 

.31923 

.94768 

.33573 

.94190 

58 

38 

.26948 

.90301 

.28625 

.95816 

.30292  .95301 

.31951 

.94758 

.33600 

.94186 

'.'.. 

30 

.26976  .96293 

.28652 

.95H07 

.303201.95293 

.31979 

.94749 

.33627 

.94176 

2] 

40 

.27004  .96285, 

.28680 

.95799 

.30348  .95284 

.32000 

.94740 

i  .33655 

.94167 

3C 

41 

.27032 

.96277 

.28708 

.95791 

.30376  .95275 

.32034 

.94730 

.33682 

.94157 

fl 

42 

.27060  .96269 

.28736 

.95782 

.30403  .95266! 

.32061 

.94721 

.a3710 

.94147 

18 

43 

.27088  .96201 

.28764 

.95774 

.30431  .95257 

.32089 

.94712 

.33737 

.94137 

K 

44 

.27116 

.96253 

.28792 

.95766 

.30459  .95248 

.32116 

.94702 

.33764 

.94127 

](» 

45 

.27144 

.96246 

.95757 

.304861.  95240  i 

.32144 

.94693 

.33792 

.94118  15 

46 

.27172 

.96238 

'28847 

.95749 

.30514  .95231 

.32171 

.94684 

.33819 

.94108  14 

47 

.27200 

.96230 

!28875 

.95740 

.30542!.  95222 

.32199 

.94674 

.33846 

.94098 

18 

48 

.27228 

.96222 

.28903 

.95732 

.30570 

.95213 

.32227 

.94665 

;.  33874  1.  94088 

12 

49 

.27256 

.96214 

.28931 

.95724 

.30597 

.95204 

.32254 

.94656 

33901  .94078 

11 

50 

.27284 

.96206; 

.28959 

.95715 

.30625 

.95195 

.82282 

.94640 

,.33929 

.94068 

10 

51 

.27312 

.96198 

.28987 

.95707 

.30653 

.95186 

.32309 

.94637 

1  .33956 

.94058 

9 

52 

.27340 

.96190 

.29015 

.95698 

:  .30680 

.95177 

.32337 

.94627 

.33983 

.94049 

8 

53 

.27368 

.96182 

.29042 

.95690 

]  .30708 

.95168 

.32364 

.94618 

.34011 

.94039 

7 

54 

.27396  .96174 

.29070 

.95681 

.30736 

.95159 

.32392 

.94609 

.34038 

.94029 

B 

55 

.27424  .96166 

.29098 

.95073 

.30763 

.95150 

.32419 

.94599 

.34065 

.94019 

5 

56 

.27452  .96158 

.29126 

.95664 

.30791 

.95142' 

.32447 

.94590 

.34093 

.94009 

4 

57 

.27480 

.96150 

.29154 

.95656 

.30819 

.95133 

.32474 

.93580 

.34120 

.93999 

8 

58 

.27508 

.96142 

.29182 

.95647 

.30846 

.95124 

.32502 

.94571 

.34147 

.93989 

2 

59 

.27536  .96134 

.29209 

.95639 

.30874 

.95115 

.32529 

.94561 

.34175  .939791  1 

60 

.27564  .96126: 

.29237 

.95630 

.30902 

.95106; 

.32557 

.94552 

.34202  .93969 

0 

/ 

Cosin  Sine  ' 

Cosin 

"Sine 

Cosin  Sine  Cosin 

Sine 

Cosin 

Sine 

/ 

74°   i 

73° 

72° 

71° 

to° 

34: 


TABLE  XX.— SINES  AND   COSINES. 


20° 

21° 

22°   ||    23° 

24° 

Sine 

Cosin 

Sine  Cosin 

,  Siue  Cosin 

Sine  Cosin 

Sine 

Cosin 

/ 

0 

.84302 

.93969 

.35837  ~.9#;5rt  :  .374(11  ~  927  18 

739073  792050 

.40674 

T  91  355 

60 

1 

.34229 

.93959 

.35804  ,9ms  .374SS  .92707 

.39100  .92039! 

.40700 

.91343 

59 

2  !  .34257 

.93949 

.35891  .93837  .37515  .92097 

.39127  .92028 

.40727 

.91331 

5;? 

3  .34284 

.93939 

.35918  .93337:!.  37543 

.92080 

.39153  .92010 

.40753 

.91319 

57 

4  .34311 

.93929 

.35945  .9:3310 

.37509 

.92675 

.39180 

.93005: 

.40780 

.91307 

56 

5 

.34339 

.93919 

.35973  .93306 

.37595 

.92604 

.39207  .91994! 

.40800 

.9121)5 

55 

6 

.34366  .93909 

.36000  .'.>:.!•,*.  C> 

.37022 

.92053 

.39234  .91982 

.40833 

.91283 

54 

y 

.34393  .9:3899 

.36027  .'.i:.5->5 

.37649 

.92042 

.39200  .91971 

.40800 

.91272 

53 

8 

.34421 

.93889 

.36054  .93274 

.37076 

.92031 

.39287  .91959 

.40;  Hi 

.91260 

52 

9 

.34448 

.93879 

.36081 

.93204 

.37703 

.92620 

.39314 

.91948 

.4001.3 

.91248 

51 

10 

.34475 

.93809 

.36108 

.9325:3 

.37730 

.92009 

.39341 

.91936 

.40939 

.91230 

50 

11 

.  34503 

.93859 

.36135 

.93243 

.37757 

.92598 

.39367 

.91925 

.40900  .!M-::i 

40 

12 

.34530 

.9:3849 

.30102 

.93232 

.37784 

.92587 

.39394 

.91914 

.40992 

.91212 

48 

13 

.34557 

.9:3839 

!  36190 

.93222 

.37811 

.92570 

.39421 

.91902 

.41019 

.01000 

14 

.34584 

.93829 

.30217 

.93211 

.37838 

.92505 

.39448 

.91891 

.41045 

.911FH 

HO 

15 

.34012 

.93819 

.36244 

.93201 

.37865 

.92554 

.39474 

.91879 

.41072 

.91176 

46 

16 

.31639 

.93809 

.36271 

.93190 

.37892 

.92543 

.39501 

.91868 

.41098 

.91104 

44 

17 

.34000 

.93799 

.36298 

.93180 

.37919  .92532 

.39528 

.91856 

.41125 

.0115.' 

4-3 

18  .34694 

19  .34721 

.93789 
.93779 

.30325  .93169 
.363521.93159 

.37940  .92521 
.37973  .92510 

.39555 
.39581 

.91845 
.918:33 

.41151 

.41178 

.91140 
91128 

42 
41 

20  .34748 

.93709 

,3637'9 

.93148 

.37999  .92499 

.39608 

.91822 

.41204 

.91116 

40 

21  .34775 

.93750 

.36406 

.93137 

.38026  .92488 

.39635 

.91810 

.41231 

.91104 

39 

22  .34803 

.93748 

.36434 

.93127 

.38053 

.92477 

.39661 

.91799 

.41257 

.91092 

38 

23  .34830 

.937:38 

.36461 

.93116 

.38080 

.92466 

.39688 

.91787 

.41284 

37 

24  .34857 

.93728 

.36488 

.93106 

.38107 

.92455 

.39715 

.91775 

.41310 

.91068 

25  .34884 

.93718 

.36515 

.93095 

.38134 

.92444 

.39741 

.91764 

.41337 

.91056 

35 

26 

.34912 

.93708 

.36542 

.93084 

.38101 

.92432 

.39768 

.91752 

.41363 

.91044 

34 

27  .34939 

.93698 

.  36569  !.9307'4 

.38188  .92421 

.39795 

.91741 

.41390 

.91032 

33 

28  .34966 

.93688 

.36596  .93063 

.38215  .92410 

.39822 

.01721' 

.41416 

32 

29  .34993 

.93677 

.36623  .93052 

.£8241 

.92399 

.39848 

.91718 

1  1  !  i  H  IN 

31 

30 

.35021 

.93667 

.36650 

.93042 

.38208 

.92388 

.39875 

.1,1706 

.41469 

!  90998 

30 

31 

.35048 

.93657 

.36677 

.93031 

.38295 

.92377 

.39902 

01004 

.41496 

.90984 

29 

32 

.35075 

.93647 

.30704 

.93020 

.38322  .92366 

.399.28  .H08 

.41522 

.90!  »72 

38 

33 

.35102 

.93637 

.36731 

.93010 

.  38349  :.  92355 

.89955 

.91671 

.41549 

34 

.35130 

.93626 

.30758 

.92999 

.38376  .92343 

:  .39982 

.91660 

.415751.90948 

26 

35 

.35157 

.93616 

.36785 

.92988 

.384031.92332 

i  .40008 

.91648 

.41602 

.90930 

25 

36 

.35184 

.93606 

.36812 

.92978 

.38430 

.92321 

.40035 

.91630 

.41628 

.90924 

24 

37 

.35211 

.  93596 

.36839 

.92967 

.38456 

.92310 

.40062 

!  91(125 

.41655 

.90911 

23 

38 

.35239 

.93585 

.36867 

.92956 

.38483 

.92299 

.40088 

.9161=3 

.41681 

.90899 

22 

39 

.35266 

.93575 

.36894  .92945 

.38510  .92287 

.40115 

.91601 

.417071.  90887 

21 

40 

.35293 

.93565 

.369211.92935 

.38537 

.92276 

.40141 

.91590 

.41734 

.90875 

20 

41 

.35320 

.93555 

.36948  .9292-1 

.38564 

.92265 

.40168 

.91578 

.41760 

.90863 

10 

42 

.35347 

.93544 

.36975 

.92913 

.38591 

.92254 

.40195 

.91566 

.41787 

.00851 

18 

43 

.35375 

.93534 

.37002 

.92902 

.38617 

.92243 

.40221 

.91555 

.41813 

.90S39 

17 

44 

.35402 

.93524 

.370291.92892 

.38044 

.92231 

.40248 

.91543 

.41840 

;;i  !>:_>i; 

16 

45 

.35429 

.93514 

.  371)56  :.  92881 

.38071 

.92220 

.4027'5 

.91531 

.41806 

/90814 

15 

46 

.35456 

.93503 

.37083  .92870 

.38098'  92209! 

.40:301 

91519 

41892 

14 

47 

.35484 

.93493 

.371101.92859 

.387251.92198! 

.40328 

.91508 

.41919 

!il(/7'90 

13 

48 

.35511 

.93483 

.37137  .92849 

.88752 

.92186 

.40355 

.91490 

.41945 

.  90778 

12 

49 

.35538 

.93472 

.37164  .92838 

.38778  .92175 

.40381 

.91484 

.41972 

11 

50 

.35565 

.93462  .37191 

.92827 

.38805  .92164'  .40408  .91472 

.41998 

!  90758 

10 

1  51 

.35592 

.93452  .37218 

.92816 

.38832  .92152 

.40434  .91461 

.  42024  !.  90741 

9 

52 

.35619 

.93441 

.37245 

.92805 

.388.7,)  .92141 

.40461  .91449 

.42051 

.90729 

8 

53 

.35647 

.93431 

.37272 

.92794 

.38886!.  92130 

.40488  .91437 

.42077 

.90717 

7 

54 

.35674 

.93420 

.37299 

.92784 

.38912  .92119 

.40514  .91425 

.42104 

.90704 

6 

55 

.35701 

.93410 

.37326 

.92773 

.38939  .92107 

.  40541  '.  91414  .42130 

.90082 

5 

56 
57 
58 

.35728 
.35755 
.35782 

.93400 
.93389 
.93379 

.37353 
.37380 
.37407 

.92762 
.92751 
.92740 

.389661.92096  .40567  .91402 
.38993  .92085  !  .40594  .91390 
.39020  .92073  .40621  .91378 

.42156 
.42183 

.42209 

.90680  4 
.90668  3 
.90655  2 

59 

.35810  .93368 

.37434 

.92729 

.39046  .92062  .40647  .91366  .42235 

.90643 

1 

60 

.35837 

.93358 

.37461 

.92718 

.31)073  .92050  .40074  .91355 

.42262 

.90631 

0 

Cosin 

Sine 

Cosin 

Sine 

Cosin  Sine 

Cosin  Sine 

Cosin 

Sine 

69° 

68° 

67°   il   666   II   65° 

TA11LE   XX. -SINES  AND  COSINES. 


343 


25° 

26° 

27° 

28° 

29° 

/ 

Sine  jCosin 

Sine 

Cosin 

Sine 

Cosin 

Sine  I  Cosin 

Sine  Cosin 

0 

742262  790631 

.43837 

.89879 

.45399 

789101 

74694  ?!  788295 

748481 

.87462 

60 

1 

.42288!.  9061  8 

.43863 

.89867 

.45425 

.89087 

.469731.88281 

.48506 

.87448 

59 

2 

.42315  .90606  i  .43889 

.89854 

.45451 

.89074 

.46999|.  88267 

.48532 

.87434 

53 

3 

.42341 

.90594 

.43916 

.89841 

.45477 

.83061 

.470241.88254 

.48557 

.87420 

57 

4 

.42367 

.90582 

.43942 

.89828 

.45503 

.89048 

.47050 

.K32-10 

.48583 

.87406 

56 

5 

.42394 

.90569 

.43968 

.89816 

.45529 

.89035 

.47076 

.88226 

.48608 

.87391 

55 

6 

.42420 

.90557 

.43994 

.89803 

.45554 

.89021 

.47101 

.88213 

.48634 

.87377 

54 

.42446  '.90545 

.44020 

.89790 

.45580 

.89008 

.47127 

.88199 

.48659 

.87363 

53 

g 

.424731.90532 

.44046 

.89777 

.45600 

.83995 

.47153 

.88185 

.48684 

.87349 

52 

9 

.42499 

.90520 

.44072 

.89764 

.45632 

.83981 

.47178 

.88172 

.48710 

.87335 

51 

10 

.42525 

.90507 

.44098 

.89752 

.45658 

.88968 

.47204 

.88158 

.48735 

.87321 

50 

11 

.42552 

.90495 

.44124 

.89739 

.45684 

.88955 

.47229 

.88144 

.48761 

.87306 

49 

12 

.42578 

.90483 

.441511.89726 

.45710 

.88942 

.47255 

.83130 

.48786 

.87292 

48 

13 

.42604 

.90470 

.44177 

.89713 

.45736 

.88928 

.47281 

.88117 

.48811 

.87278 

47 

14 

.42631 

.90458 

.44203 

.89700 

.45762 

.88915 

.47306 

.88103 

.48837 

.87264 

46 

15 

.42657 

.90446 

.44229 

.89687 

.45787 

.88902 

.47332 

.88089 

.48862 

.87250 

45 

16 

.42683 

.90433 

.44255 

.89674 

.45813 

.88888 

.47358 

.88075 

.48888 

.87235 

44 

17 

.42709 

.90421 

.44281 

.89662 

.45839 

.88875 

.47383 

.88062 

.48913 

.87221 

43 

18 

.42736 

.90408 

.44307 

.89649 

.45865 

.88862 

.47409 

.88048 

.48938 

.87207 

42 

19  .42762 

.90396 

.44333 

.89636 

.45891 

.88848 

.47434 

.88034 

.48964 

.87193 

41 

20 

.42788 

.90383 

.44359 

.89623 

.45917 

.88835 

.47460 

.88020 

.48989 

.87178 

40 

21 

.42815 

.90371 

.44385 

.89610 

.45942 

.88822 

.47486 

.88006 

.49014 

.87164 

£9 

22 

.42841 

.90358 

.44411 

.89597: 

.45968 

.88808 

.47511 

.87993 

.49040 

.87150 

*'O 

23  1  .42867 

.90346 

.44437 

.89584 

.45994 

.88795 

.47537 

.87979 

.49065 

87136 

37 

24  .42894 

.90334 

.44464 

.89571  ! 

.46020 

.88782 

.47562 

.87965 

.49090 

.87121 

36 

25  .42920 

.90321 

.44490 

.89558: 

.46046 

.88768 

.47588 

.87951 

.49116 

.87107 

35 

26 

.42946 

.90309 

.44516 

.89545 

.46072 

.88755 

.47014 

.87937 

.49141 

.87093 

34 

27 

.42972 

.90296 

.44542 

.89532 

.46097 

.88741 

.47039 

.87923 

.49166 

.87079 

33 

28 

.42999 

.90284 

.44568 

.89519 

.46123 

.88728 

.47665 

.87909 

.49192 

.87064 

32 

29 

.43025 

.9^271 

.44594 

.89506 

.46149 

.88715 

.47690 

.87896 

.49217 

.87050 

31 

30 

.43051 

.90259 

.44620 

.89493 

.46175 

.88701 

.47716 

.87882 

.49242 

.87036 

30 

31 

.43077 

.90246 

.44646 

.89480 

.46201 

.88688 

.47741 

.87868 

.49268 

.87021 

29 

32 

.43104 

.90233 

.446721.89467 

.46226 

.88674 

.47767 

.87854 

.49293 

.87007 

28 

33 

.43130 

.90221 

.44698 

.89454 

.40252 

.88661 

.47793 

.87840 

.49318 

.86993 

27 

34 

.43156 

.90203; 

.44724 

.89441 

.46278 

.88647 

.47818 

.87826 

.49344 

.86978 

2G 

35 

.43182 

.90196: 

.44750 

.89428 

.46304 

.8863-1 

.47844 

.87312 

.49369 

.86964 

25 

36 

.43209 

.90183 

.44776 

.89415 

.46330 

.88620 

.47869 

.87798 

.49394 

.86949 

24 

37 

.43235 

.90171 

.44802 

.89402 

.46355 

.88607 

.478951.87784 

.49419 

.86935 

23 

38 

.43261 

.90153 

.44828 

.89389 

.46381 

.88593 

.47920 

.87770 

.49445 

.86921 

23 

39 

.43287  .90146 

.44854 

.80376 

.46407 

.88580 

.47946 

.87756 

.49470 

.80906 

21 

40 

.43313 

.90133 

.44880 

.89363 

.46433 

.88566 

.47971 

.87743 

.43495 

.86892 

20 

41 

.4&340 

.90120 

.44906 

.89350 

.46458 

.88553 

.47997 

.87729 

.49521 

.86878 

10 

42 

.43386  .90103 

.449321.89337 

.46484 

.88539 

.48022 

.87715 

.49546 

.86863 

lu  i 

43 

.43392  .90035 

.44958  .89364 

.46510 

.88526 

.48048 

.87701 

.49571 

.86849 

17 

44 

.43418 

.90-082 

.44334  .89311 

.46536 

.88512 

.480731.87687 

.49590 

.86834 

16 

45 

.4:3445 

.90070 

.45010  .89298 

.46561 

.83499 

.480991.87673 

.49622 

.86820 

15 

46 

.43471 

.90057 

.45036 

.89285 

.46587 

.88485 

.481241.87659 

.49647 

.86805 

14  j 

47 

.43497 

.90045 

.45082 

.89272 

.46613 

.88472 

.48150 

.87645 

.49672 

.86791 

13  i 

48 

.43523 

.900-52 

.45088 

.89259 

.46639 

.88458 

.48175 

.87631 

49697 

.86777 

12  i 

49 

.43549 

.90019 

.45114 

.89245 

.46664 

.88445 

.48201 

.87617 

.49723 

.86762 

11 

50 

.43575 

.90007 

.45140 

.89232 

.46690 

.88431 

.48226 

.87603 

.49748 

.86748 

10 

51 

.43602 

.89994 

.45166 

.89219 

.46716 

.88417 

.48252 

.87589 

.49773 

.86733 

r) 

52 

.43628 

.801)31 

.45192 

.89203 

.46742 

.88404 

.48277 

.87575 

.49798 

.86719 

8 

53 

.43654 

.89968 

.45218 

.89193 

.46767 

.88390 

.48303 

.87561 

.49884 

.86704 

7 

54 

.43680 

.89956 

.45-243 

.89180 

.46793 

.88377 

.48328 

.87546 

.49849 

.86690 

C 

55 

.43706 

.891)43 

.  45269 

.89167 

.46819 

.88363 

.48354 

.87532 

.49874 

.86675 

5 

56 

.43733 

.89930 

.45295 

.89153 

.46844 

.88349 

.48379 

.87518 

.49899 

.86661 

4 

57 

.43759 

.89918 

.45321 

.89140 

.46870 

.88336 

.48405 

.87504:1.49924 

.86646 

3 

58 

.437&5 

.89905 

.45347  .89127 

.46896 

.88322 

.48430 

.87490 

.49950 

.86632 

2 

59 

.43811 

.89892 

.453731.89114 

.46921 

.88308 

.48456 

.87476 

.49975 

.86617 

1 

60 

.48837  .89879 

.45399 

.89101 

.46947 

.88295 

.48481 

.67462 

.50000 

.86603 

_0 

/ 

Cosin  |  Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

/ 

64°   ll   GDa    i   C2° 

6iu   II  ,  60° 

TABLE    XX.— SIXES  AND  COSINES. 


30* 

31°   |1    32° 

qoo 
OO 

34' 

Sine 

Cosin 

Sine 

Cosin 

Sine  Cosin  Sine  1  Cosin 

Sine  j  Cosin 

f 

0 

750000 

.86003 

.51504 

.85717 

752992  784K05  7544(54  ~.  83807 

.55919  '.K2901 

60 

1 

.50025 

.86588 

.51529 

.85702 

.53017  .84789  .54488  .83851 

.55943  .82887 

69 

2 

.50050 

.86573 

.51554 

.85687 

.530411.84774  .64513  .83835 

.55908  .82871 

58 

8 

.50076 

.86559 

.51579 

.85672 

.53066  1.84759  ;  .54537  .83819 

.55992  L  82855 

57 

4 

.50101 

.86544 

.51604 

.85657 

.530911.84743  .51561 

.83804 

.56016 

.82839 

56 

5 

.50126 

.86530 

.51628 

.85642 

.53115  .84728 

.54586 

.83788 

.56040 

.82822 

55 

6 

.50151 

.86515 

.51653 

.85627 

.53140  .84712  !  .546101.83772 

.56064 

.S2S06 

54 

? 

.50176 

.86501 

.51678 

.85612 

,53164 

.84697  .546351.83756 

.50088 

.S2',yo 

53 

8 

.50201 

.86486 

.51703 

.85597 

.53189 

.84681  .54659  .83740 

.56112 

.8-2773 

53 

9 

.80827 

.86471 

.51728 

.85582 

.63214 

.84666  i  .546831.83724 

.56136 

.82757 

51 

10 

.50252 

.86457 

.51753 

.85567 

.53238  .84650  .54708  .837u8 

.50160 

.82741  50 

11 

.50277 

.86442 

.51778 

.85551 

.53263 

.84635"  .54732  .83692 

.561  SI 

.82724 

49 

12 

.50302 

.86427 

.51803 

.85536 

.532881.84619 

.54756 

.83'.j;y 

.50208 

.82708 

48 

13 

.50327 

.86413 

.51823 

.85521 

.53312  .84604 

.54781 

.83600 

.56232 

.82092 

47 

14 

.50352 

.86398 

.51852 

.85506 

.53337 

.84588 

.54805 

.83645 

.56256 

.82675 

46 

15 

.50377 

.86384 

.51877 

.85491 

.53361 

.84573 

.54829  .83629 

.56280 

.82059 

45 

16 

.50403 

.86369 

.51902 

.85476 

.53386 

.84557 

.54854  .83613 

.56305 

.82643 

44 

17 

.50428 

.86:354 

.51927 

.85461 

.53411 

.84542 

54878  83597 

.56329 

82626 

43 

18 

.50453 

.86340 

.51952 

.85446 

.53435 

.84526 

.54902  .83581 

.56353 

.82610 

42 

19 

.50478 

.86325 

.51977 

.85431 

.53460!.84311 

..'.4927 

.83365 

.56377 

.82593 

41 

20 

.50503 

.86310 

.52002 

.85416 

.53484 

.84495 

.54951  .83549 

.56401 

.82577 

40 

21 

.50528 

.86295 

.52026 

.85401 

.53509 

.84480 

.54975  .83533 

.56425 

.82561 

39 

22  !  .50553 

.86281 

.52051 

.85385 

.533341.84464 

.54990 

.83517 

.56449 

.82344 

33 

23  .50578 

.86266 

.52076 

.85370 

.53558 

.84448 

.5502-4 

.83501 

.56473 

.82528 

37 

24 

.50603 

.86251 

.52101 

.85355 

.53583 

.84433 

.55048 

.83485 

.56497 

.8251  1 

3G 

25 

.50628 

.86237 

.52126 

.85340 

.53607 

.84417 

.55072 

.83469 

.56521 

.82495 

85 

26 

.50654 

.86222 

.52151 

.85325 

.53632 

.84402 

.55097 

.83453 

.56545 

.82478 

34 

Of"» 

.50679 

.86207 

.52175 

.85310 

.53656 

.84386 

.55121 

.83437 

.56569 

.82462 

83 

28 

.50704 

.86192 

.52200 

.85294 

.53681 

.84370 

.55145 

.83421 

.56593 

.82446 

32 

29 

.50729 

.83178 

.52225 

.85279 

.53705 

.84355 

.55169 

.83405 

.56617 

.82S29 

81 

30 

.50754 

.86163 

.52250 

.85264 

.53730 

.84339  .55194 

.83389 

.56641 

.82413 

30 

31 

.50779 

.86148 

.52275 

.85249 

.53754 

.84324  .55218 

.83373 

.56665 

.82396 

!29 

32 

.50804 

.86133 

.5.3299 

.85234 

.53779 

.84308  .55243:.  8=3356  .56689 

.82380 

28 

33 

.50829 

.86119 

.53324 

.85218 

.53804 

.84292  .P5266 

.83340  i  .56713 

.82363 

27 

34 

.50854 

.86104 

.52349 

.85203 

.53828 

.84277 

.55291 

.83321  !  .56736 

.82347 

26 

35 

.50879 

.86089 

.52374 

.85188 

.53853  \.  84261 

.55315 

.83308  i  .56760 

.82330 

25 

36 

.50904 

.86074 

.52399 

.85173 

.53877 

.84245 

.55339 

.83292  !  .56784 

.8231-1 

24 

37 

.50929 

.86059 

.52423 

.85157 

.53902 

.84230 

.55363 

.83276  !  .56808 

.82297 

23 

38 

.50954 

.86045 

.52448 

.85142] 

.53926 

.84214 

.55388 

.83200  .56832 

.82281 

22 

39 

.50979 

.860301 

.52473 

.85127 

.53951 

.84198 

.55412  .83244  .56856 

.82204 

21 

40 

.51004 

86015  1 

.52498 

.85112 

.53975 

.  84182 

.55436 

.83228 

.50880 

.82248 

20 

41 

.51029 

.860001 

.52522 

.85096 

.54000 

.84167! 

.55460 

.83212  h  .56904 

.82231 

19 

42 

.51054 

.  859::15 

.52547 

.85081 

.54024 

.84151 

.55484 

.83195:  .56928 

.82214 

18 

43 

.510791.85370 

.52572 

.85066 

.54049 

.84135 

.55509 

.83179  .56952 

82198 

17 

44 

.51104 

.85356 

,52597 

.85051 

.54073 

.84120 

.55533  j  .83163  '.  '•  .50976  i  .82181 

16 

45 

.51129 

.85941 

.52621 

.85035 

.54097 

.84104 

.55557  .83147:  .57000 

.82165 

15 

46 

.51154 

.85926 

.52646 

.85020 

.54122 

.84088 

.55581  .83131  .57024 

.82148 

14 

47 

.51179 

.85911 

.53671 

.85005 

.54146 

.84072 

.55605 

.83115;  .57047 

.82132 

13 

48 

.51204  .85896 

.52696 

.84989 

.54171 

.84057 

.55630 

.83098  .57071 

.82115 

12 

49 

.51229|.85881 

.52?20 

.  84974  i 

.54195 

.84041 

.55654 

.  83082  ii  .57095 

.82098 

11 

50 

.51254  .85866 

.52745 

.84959 

.54220 

.8-1025 

.55078 

.83006  .57119 

.82082 

10 

51 

.512791.85851 

.52770 

.84943 

.54244 

.84009! 

.55702 

.83050  .57143 

.82065 

9 

52 

.  51304 

.85836 

.53794 

.84928 

.542691.83994 

.55726 

.83034  .571(57 

.82048 

8 

53 

.51329 

.85821 

.52819 

.84913 

.54293  |.83978 

.55750 

.83017,;.  57191 

.82032 

7 

54 

.51354 

.85806 

.52844 

.84897 

.54317.88962  .55775 

.83001!  .57015 

.82015 

;  6 

55 

.51379 

.85792 

.52869 

.84882 

.  54342  1.83946  .55799 

.82985'  .57238 

.81999 

5 

56 

.51404 

.85777 

.52893 

.84866 

.543661.83930  .55823 

.82969  .57262 

.81982 

4 

57 

.51429 

.85762 

.52918 

.84851 

.54391  |.  83915  .55847 

.82953-  .57286 

.81905 

3 

58 

.51454 

.85747 

.52943 

.84836! 

,64415.88899  .55871 

.82936,  .57310 

.81949 

2 

59 

.51479 

.85732 

.52967 

.84820 

.54440  .83883  .53895 

.82920  ,57334 

.si  >»:;•_' 

1 

60 

.51504 

.85717 

.52992 

.84805 

.54464  .  8:3867  Ij  .55919 

.82904  .57358 

.81915 

0 

/ 

Cosin  Sine 

Cosin 

Sine 

Cosin  |  Sine  Cosin 

Sine 

Cosin 

Sine 

i 

59° 

58° 

57°       56° 

55° 

TABLE    XX.— SINES   AND   COSINES. 


345 


35°   !  |    36° 

37°    H    38°    ||    39° 

/ 

Sine  i  Cosin 

Sine  ICosin 

Sine 

Cosin 

Sine  Cosin  !|  Sine 

Cosin 

~o 

.  57358  LsiiUo  .58779  .80902 

.60182 

•79864 

.61566  .78801 

.62932 

.77715  60 

1 

.57381  .81899  .58802 

.80885 

.60205 

.79846 

.61589  .78783 

.62955 

.77696!  59 

2 

.57405  :.M8AJ  .58826  .80867 

.60228 

.79829 

.61612  .78765 

.62977 

.77678)  58 

3 

.57429  .81  865  !  .58849  .80850! 

.60251 

.79811 

.61635  .78747 

.63000 

.77660  57 

4 

.57453  .81848 

.58873  .80833 

.60274 

.79793 

.61658  .78729! 

.63022 

.77641 

56 

5 

.57477  .  81832  i 

.58896 

.80816 

.60298  .79776 

.61681 

.78711 

.63045 

.77623 

56 

C 

.57501 
.57524 

.81815| 
.81798 

.58920 
.58943 

.80799' 

.80782 

.60321  .79758 
.  60344  S.  79741 

.61704 
.61726 

.78694 

.78676J 

.63068 
.63090 

.77605  54 
.77586  53 

8 

.57548  .81782 

.58967 

.80765 

.60367 

.79723 

.61749 

.78658J 

.63113 

.77568 

52 

9 

.57572 

.81765 

.58990  .80748! 

.60390 

.79706  .61772 

.78640! 

.63135 

.77550'  51 

10 

.57596 

.8ir48| 

.59014  .80730; 

.60414 

.79688.  .61795 

.78622 

.63158 

.77531 

50 

11 

.57619 

.81731 

.59037 

.80713 

.60437 

.  79671  !  .61818 

.7'8604 

.63180 

.77513 

40 

12 

.57643 

.81714 

.59061 

.80696 

.60400 

.79653 

.61841 

.78586 

.63203 

.77494 

18 

13 

.57667 

.81698 

.59084  .80679 

.60483 

.79635 

.01864 

.7'8568 

.63225  .77476 

47 

14 

.57691 

.81681 

.591081.80662 

.60506 

.79618 

.61887 

.78550 

.63248  .77458  46 

15 

.57715 

.81664 

.59131 

.80644 

.60529 

.79600  i  .61909  .78532, 

.63271  .77439  45 

16 

.57738 

.81647 

.59154 

.80627i 

.60553 

.79583  .61932  .78514 

.63293  .77421  44 

17 

.57762 

.81631 

.59178 

.80610! 

.60576 

.79565  .61955!.  78496 

.63316  .77402  43 

18 

.57786 

.81614 

.59201 

.80593' 

.60309 

.79547 

.61978  .78478 

.63338  .77384'  42 

10 

.57810 

.81597 

.59225 

.80576 

.60622 

.79530  .62001  .78460 

.633611.77366  41 

20 

.57833 

.81580 

.59248 

.80558; 

.60645 

.79512  .62024  .  78442  | 

.63383  .77347 

40 

21 

.57857 

.  81563  ! 

.59272 

.  80541 

.60668 

.  79494  p  .62046  .78424 

.63406  .77329J  39 

oo 

.57881 

.81546 

.59295 

.80524 

.60691 

.79477  .  .62069  .78405 

.63428  .77310  38 

23 

.57904  .81530 

.59318 

.80507 

.60714 

.79459  .62092  .78387' 

.63451  .77292  37 

24 

.57928 

.81513 

.59342 

.80489 

.60738 

.79441  .62115  .788691 

.  63473  !.  77273!  36 

25 

.57952  .81496 

.59365 

.80472 

.60761 

.79424  .62138 

.78351! 

.63496 

.77255  35 

26 

.57976  .81479 

.59389 

.80455 

.60784 

.79406  .62160 

.78333 

.63518 

.77236^34 

27 

.57999 

.81462 

.59412 

.80438 

.60807 

.79388!  .62183  .78315 

.63540  .77218!  33 

28 

.58023 

.81445; 

.59436  .80420 

.60830  .79371!  .62206  .78297 

.63563 

.77199  32 

29 

.58047 

.81428 

.59459  .80403 

60853  .79353  .62229 

.78279! 

.63585 

.77181  !  31 

30 

.58070  .81412 

.59482  .80386; 

.60876 

.79335 

.62251 

.78261 

.63608 

.77162  30 

31 

.580941.81395 

.59506  .80368 

.60899 

.79318  .62274 

.78243; 

.63630 

.77144  29 

32 

.58118!.  81378  i 

.593291.80351 

.60922  .793001!  .62297 

.78225 

.63653  .77125  28 

33 

.581411.81361 

.595521.80334 

.60945 

.79282  .62320 

.78206 

.63675 

.77107  27 

34  1.581  65  i.  813441 

.59576 

.80316 

.60968 

.79264  !  .62342 

.78188 

.63698  .77088  26 

35 

.58189;  .  81327 

.59599 

.80299! 

.60991 

.79247  .62365  .78170 

.63720  .77070  25 

36 

.58212  .81310;  .59622 

.802821 

.61015 

.79229  i  .62388  .78152 

.63742 

.77051  24 

37 

.58236.81293  .59646 

.80264 

.61038 

.79211!!  .63411 

.78134; 

.63765 

.77033 

23 

38 

.58260  .81276 

.59069 

.80247 

.61061 

.79193  .62433 

.78116! 

.63787 

.77014  22 

39  .58283  .81259 

.59693 

.80230 

.61084 

.79176 

.62456 

.78098 

.63810 

.76996!  21 

40 

.  58307  i.  81242 

.59716 

.80212: 

.61107 

.79158 

.62479 

.78079 

.63832 

.76977  20 

41 

.58-330  '.81225  .59739 

.80195  I  .61130 

.79140  .62502 

.78061 

.63854 

.76959  19 

42 

.58334  .81208  .59763 

.80178 

.61153 

.79122  .625241.780431 

.63877  .76940  18 

43 

.58378  .81191 

.59786 

.80160 

.61176 

.79105 

.62547 

.78025 

.63899  .76921  17 

44 

.58401  >.81  174! 

.59809 

.80143 

.61199 

.79087 

j  .62570 

.78007 

.639221.76903!  16 

45 

.58425  .81157 

.59832 

.80125 

.61222 

.79069 

'.62592  .77988! 

.63944 

.7'6884  15 

46 

.58449  .811401 

.59856 

.80108 

.61245 

.79051 

.62615 

.77970 

.63966  .76866  14 

47 

.58472,.  81  123  S  .59879 

.80091 

.61268 

.79033 

.62638  .  77952  ! 

.63989  .76847  13 

48 

.58496 

.81106 

.59902 

.80073 

.61291 

.79016  .62660 

.  77934  ' 

.64011  .76828  12 

49 

.58519 

.81089 

.59926 

.  80056  ! 

.61314 

.78998  .62683 

.77916 

.64033 

.768101  11 

50 

.58543 

.81072 

.59949 

.80038 

.61337 

.  78980  i  .62706 

.77897 

.64056 

.76791 

10 

51 

.58567 

.81055 

.59972 

.80021! 

.61360 

.78982  .62728 

.77879 

.64078 

.76772 

9 

52 

.58590 

.81038' 

.59995 

.80003 

.61383 

.  78944  il  .62751 

.77861  ! 

.64100 

.76754 

8 

53 

.58614 

.81021i 

.60019 

.79986 

.61406 

.78926 

.62774 

.778431 

.64123 

.76735 

54 

.58637 

.81004 

.60042 

.79968 

.61429 

.78908 

.62796 

.77824 

.64145 

.70717 

6 

55 

.58661 

.80987 

.60065  .79951 

.61451 

.78891 

.62819 

.77806 

.64167 

.76698  5 

56 

.58684 

.  80970  .60089 

.79934 

.61474 

.78873 

.62842 

.77788 

.64190 

.76679  4 

57 

.58708 

.80953  .60112  .79916 

.61497 

.78855 

.62864 

.77769 

.642121.76661  3 

58 

.58731 

.80936  .60135  .79899 

.61520 

.78837 

.62887 

.  77731 

.64234 

.76642 

2 

59 

.58755 

.80919!  .60158 

.  79881  i 

.61543 

.78819 

.62909 

.77733 

.64256 

.76623 

60 

.58779  .80902  1  .60182 

.79864 

.61566 

.78801 

.62932 

.77715 

.64279 

.76604 

0 

/ 

Cosin  1  Sine  Cosin  Sine 

Cosin  Sine 

Cosin 

Sine 

Cosin 

Sine 

/ 

54°   II   53° 

52° 

51° 

50° 

TABLE  XX.— SINES  AND  COSINES. 


40°       41°   1!   42° 

43° 

44° 

Sine  !  Cosin  |  Sine 

Cosin 

Sine  j  Cosin 

Sine  Cosin 

Sine  Cosin 

' 

.64279 

.76604 

.65000 

.75471 

.66913 

.74314 

.68200 

.73135 

.69406 

.71934 

60 

.61301 

.76586 

.65628 

.75452 

.66935 

.74295 

.68221 

.73116 

.69487 

.71911 

59 

.64333 

.76567 

.65650 

.75433 

.66956 

.74276 

.68242 

.73096 

.69508 

.71894 

:  53 

.64346 

.  76548  l 

.65672 

.75414 

.66978 

.74256 

.68264 

.73076 

.69529 

.71873!  57 

.64368 

.76530 

.65694 

.75395 

.66999 

.74237 

.68285 

.73056 

.69549 

.71853 

r>r> 

.64390 

.76511 

.65716 

.75375 

.67021 

.74217 

.68306 

.73036 

.69570 

.71833  55 

.64412 

.76492 

.65738 

.75356 

.67043 

.74198 

.68327 

.73016 

.69591 

.71813 

M 

.64435 

.76473 

.65759 

.75337 

.67064 

.7-1178 

.68349 

.72996 

.69612 

.71792  53 

.61457 

.76455 

.65781 

.75318 

.67086 

.74159 

.68370 

.72976 

.69633 

.71772 

i  52 

.61479 
.64501 

.76436 
.76417 

.65803 
.65825 

.75233, 
.75280 

.07107 
.67129 

.74139 
.74120 

.68391 
.68412 

.72957 
.72937 

.69654 
.69675 

.717521  51 
.71732,  50 

.64524 

.76398' 

.65847 

.75261 

.67151 

.74100 

.68434 

.72917 

.69696 

.71711 

1*9 

.  64546  !.  763301 

.653o3 

.75211 

.67172 

.7'4080 

.68455 

.72897 

.69717  .71691 

1  48 

.6  1563'.  76331  | 

.65391 

.75:i,A2 

.67194 

.74061 

.68476 

.72877 

.69737 

.71671 

!  47 

.64590  .76343 

.65913 

.75203 

.67215 

.74041 

.68497 

.72857 

.69758 

.71650 

;  46 

.64612 

.76323 

.65935 

.75184 

.67237 

.74022 

.68518 

.72837 

.69779 

.71030  45 

.64635 

.763041 

.65953 

75165 

.67258 

.74002 

.68539 

.72817 

.69800 

.71610  44 

.64657 

.76236  .65378 

75146 

.67280 

.73983 

.68561 

.72797 

.69821 

.71590 

43 

.64679 

.76267  .63000 

75128 

.67301 

.73963 

.68582 

.  72777 

.69842 

.71569 

42 

,64701 

.76248  .63022 

75107  i 

.67323 

.73944 

.68603 

.72757 

.698621.71549 

41 

.64723 

.76229  ;  .66044 

75088  .67344 

.73924 

.68624 

.72737 

.69883  .71529 

140 

.64746 

.76210  '1.66066 

75039 

.67366 

.73904 

.68645 

.72717 

.69904  .71508 

'39 

.64768  .76192  .66033 

7535J 

.67387 

.73885 

.68666 

.72697 

.69925  .71488 

38 

.64790  .76173  .63103 

75030 

.67409 

.73865 

.68688 

.72677 

.69946  .71468 

37 

.64812|.76154  .66131 

75011 

.67430 

.73846 

.68709 

.72657 

.69966  .71447 

36 

.64834  j.76135  .66153 

74332 

.67452 

.73820 

.63730 

.73637 

.699871.71427 

35 

.648561.76116 

.66175 

74373  ! 

.67473 

.73808 

.63751 

.72617 

.70008  .71407 

34 

.  64878  !.  76097: 

.66197 

74953  ! 

.67495 

.73787 

.68772 

.72597 

.700291.71386 

33 

.649011.76078 

.63318 

74331 

.67516 

.73767 

.68793 

.72577 

.700491.71366 

32 

.64923  !  76059 

.60240 

71915 

.67538 

.73747 

.68814 

.72557 

.700701.71345 

31 

.64945;.  76041 

.66262 

74896 

.67559 

.73728 

.68835 

.72537 

.70091 

.71325J  30 

.64967 

.76022 

.66284 

74876' 

.67580 

.73708 

.68857 

.72517 

.70113 

.71305!  29 

.64939  .76003  i  .68306 

74857 

.67602 

.73683 

.68878 

.7.2497 

701  32  1.71284 

28 

.650111.75984  j  .66327 

74833 

.67623 

.73669 

.6:5899 

.72477 

.70153:.  71204 

27 

.  65033  j.  75965  .66349 

74818 

.67645 

.73649 

.68920 

.72457 

.70174  .71243 

26 

.65055  .75946 

.66371 

74799  ! 

.67666 

.73629 

.68941 

.  72437  ; 

.70195 

.71223 

35 

.  65077  j.  75927 

.6(3393 

74780 

.67688 

.73610 

.68962 

.72417 

.70215 

.71203 

24 

.  65100  .  75903  .66414 

74760 

.67709 

73590 

.68983 

.72397' 

.702361.71182 

88 

.65122  .75383  .66436 

74741 

.67730 

73570 

.69004 

.72377 

.70257  .71162 

22 

.65144  .75870  .60  158 

74722 

.67752 

73551 

.69025 

.72357  .  70277  i.  71  141 

21 

.65166 

.75851 

.68480 

74703  j 

.67773 

73531 

.69046 

.72337 

.70298 

.71121 

20 

.65188 

.  75832  ! 

.66501 

74683; 

.67795 

73511 

.69067 

72317 

.70319 

.71100 

19 

.65^10 

.75813 

.66523 

74664  ! 

.67816 

73191 

.69088 

72297 

.703391.71030 

18 

.65232 

.75794 

.66545 

74644;  .67837 

73172 

.69109 

72277 

.70300  .71(559 

17 

.65254 

.75775 

.68566 

74625  .67859 

.73452 

.69130 

72257 

.70381  .71039 

16 

.65276  .75756 

.66588 

74606  .67880 

.73432 

.69151 

72236 

.70401  .7101!) 

15 

.652981.75738 

.66610 

.74588  i  .67901 

.73413 

.69172 

72216  i 

.70422  .70W!-i 

14 

.65320  .75719 

.66632 

.74567  .67923 

.73393 

.69193 

72196 

.70443;.  70978 

13 

.65342  .75700 

.666531.74548  j  .67944 

.73373 

.69214 

72176 

.70463  .70957 

12 

.65364  .75680 

.66675  .74528  .67965 

.73353 

.69235 

72156 

.704H4  .roirrr 

11 

.65386 

.75661 

.66697 

.74509  .67987 

.73333 

.69256 

72136  i  .70505  .70J10 

10 

.65408 

.75642 

.66718  '.74489  .68008 

.7.3314 

.69277 

.72116  .70525  .70896 

9 

.654301.75623 

.66740!.  74470  .68029 

.73294 

.69298 

72095  .70546  .70875 

8 

.  65452  i.  75604 

.66762  .74451  !  .68051 

.73274 

.69319 

.72075  .70567  '.70355 

7 

.65474  .75585 

.66783  .74431  !  .68072 

.73254 

.69340 

72055  .70587  .70834 

6 

.65496  .75566 

.66805  .74412  !  .6,8093 

.73234  .69361  .72035  .70608  .70813 

5 

.655181.75547 

.66827  .74392  j  .68115 

.73215 

.69382  .72015  .70628..  70793 

4 

.65540  .75528 

.66848  .74373 

.68136 

.73195 

.69403 

.71995!  .706491.70772 

3 

.65562 

.75509 

.66870  .74353 

.68157 

.73175 

.  69424  .  71974  1  1  .  70670  .  70752 

2 

.65584 

.75490 

.66891  .74334 

.68179 

.73155 

.  69445  i  .  71954  .  70690  .  70731 

1 

.65606  .75471 

.66913  .74314 

.68200 

.73135 

.  69466  j.  71  934  .70711  .70711 

0 

Cosin  Sine  Cosin  Sine  Cosin 

Sine 

Cosin 

Sine  Cosin  ,  Sine 

/ 

49° 

48°       47° 

46°   !l   45°   1 

TABLE   XXI.-TANGENTS  AND  COTANGENTS. 


0° 

1°            !             2°                         3° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang  1  Cotan.Gr 

0 

.00000 

Infinite. 

.01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811 

1 

.00029 

3437.75 

.01775 

58.3506 

.03521 

28.3994 

.05270 

18.97'55 

2 

.00058 

1718.87 

.01804 

55.4115 

.03550 

28.1C64 

.05299 

18.8711 

3 

.0003? 

1145.92 

.01833 

54.5G13 

.03579 

27.9372 

.05328 

18.7678 

4 

.00116 

859.436 

.01862 

53.7086 

.03609 

27.7117 

.05357 

18.6656 

6 

.00145 

687.549 

.01091 

52.C321 

.03638 

27.4G99 

.05387 

18.5645 

0 

.0017'5 

572.957 

.01920 

52.C307  : 

.03667 

27.2715 

.05416 

18.4645 

7 

.00204 

431.106 

.01949 

51.3032 

.03696 

27.0566 

.05445 

18.3655 

8 

.00333 

429.713 

.01978 

50.5185 

.03725 

26.8150 

.0547'4 

18.2677 

!» 

.00262 

381.971 

.02007 

49.8157 

.03754 

26.6367 

.05503 

18.1708 

10 

.00291 

343.774 

.02036 

49.1039 

.03783 

26.4316 

.05533 

18.0750 

11 

.00320 

312.521 

.02066 

48.4121 

.03812 

2G.2296 

.05562 

17.9802 

12 

.00343 

JioO.47'8 

.02005 

47.7395 

.03J42 

26.0307 

.05501 

17.8863 

13!    .00313 

2G4.441 

.02124 

47.0853 

.03871 

25.8348 

.056CO 

17.7934 

14  1   .0040? 

245.552 

.02153 

46.4489 

.03900 

25.6418 

.05649 

17.7015 

18     .C043G 

229.182 

.02182 

45.8294 

.03929 

25.4517 

.05678 

17.6106 

1J,    .00-105 

214.858 

.02211 

45.2261 

.03958 

25.2644 

.05708 

17.5205 

1?J    .00^3 

5^2.213 

.02240 

44.6386 

.03987 

25.0798 

.05737 

17.4314 

18    .00524 

130.984 

.02269 

44.0661 

.04016 

24.8978 

.05766 

17.3432 

19 

.00553 

180.932 

.02298 

43.5081 

.04046 

24.7185 

.05795 

17.2558 

go 

.00583 

171.885 

.02328 

42.9641 

.04075 

24.5418 

.05824 

17.1693 

21 

.00611 

163.700 

.02357 

42.4335 

.04104 

24.3675 

.05854 

17.0837 

.  i 

.00o-10 

150.259 

.02386 

41.9158 

.04133 

24.1957 

.05883 

16.9990 

23 

.OOOJ9 

1-19.465 

.02415 

41.4106 

.04162 

24.0263 

.06012 

16.9150 

24  1   .00698 

113.237 

.02444 

40.9174 

.04191 

23.8593 

.05941 

16.8319 

85 

.00737 

137.507 

.02473 

40.4358 

.01220 

23.6945 

.05970 

16.7496 

;.-; 

.00756 

132.219 

,02502 

89.9655 

•.04250 

23.5321 

.05999 

16.6681 

27 

.00785 

127.321 

.02531 

39.5059 

.04279 

23.3718 

.06029 

16.5874 

88 

.00815 

122.774 

.02560 

39.0568 

.04308 

23.2137 

.06058 

16.5075 

;.".' 

.00844 

118.540 

.02589 

38.6177 

.04337 

23.0577 

.06087 

16.4283 

30 

.00873 

114.589 

.02619 

38.1885 

.04366 

22.9038 

.06116 

16.3499 

°1 

.00902 

110.892 

.02648 

37.7680 

.04395 

22.7519 

.06145 

16.2722 

::: 

.00^31 

107.426 

.02677 

37.3579 

.04124 

22.6020 

.C6175 

16.1952 

83 

.000  JO 

104.171 

.02706 

86.9500 

.04454 

22.4541 

.06204 

16.1190 

84 

1J1.1J7 

.02735 

3i).r,6-.7 

.04483 

22.3081 

.06233 

16.0435 

86 

.01018 

9.12179 

[02764 

3(5.1776 

.04512 

22.1640 

.06262 

15.9687 

86 

.01047 

95.4SU5 

.02793 

35.8006 

.01511 

22.0217 

.06291 

15.8945 

87 

.01076 

02.9085 

.02822 

35.4313 

.04570 

21.8813 

.06321 

15.8211 

38 

.01105 

90.4633 

.02351 

35.0695 

.04599 

21.7426 

.06350     15.7483 

39l   .01185 

8^.1436 

.02881 

34.7'im 

.016-28 

21.6056 

.06379 

15.6762 

40 

.01164 

85.9398 

.02910 

34.3678 

.04658 

21.4704 

.06408 

15.6048 

,(< 

.01193 

83.8435 

.02939 

34.0273 

.04687 

21.3369 

.06437 

15.5340 

12 

.01222 

81.8470 

.02968 

33.6935 

.04716 

21.2049 

.06467 

15.4638 

43 

.01251 

79.9434 

.02997 

a3.  3662 

.04745 

21.0747 

.06496 

15.3943 

n 

.01280 

78.1263 

.03026 

33.0452 

.04774 

20.9460 

.06525 

15.3254 

45 

.OKJOO 

76.3900 

.03055 

32.7303 

.01803 

20.8188 

.06554 

15.2571 

46 

.C1338 

74.7292 

.03084 

32.4213 

.04833 

20.6932 

.06584 

15.189'J 

47 

.01367 

73.1390 

.03114 

3->.1181 

.04862 

20.5691 

.06613 

15.1222 

48 

.01396 

71.6151 

.03113 

31.8205 

.04891 

20.4465 

.06642 

15.0557 

49 

.01425 

70.1533 

.03172 

31.5284 

.04920 

20.3253 

.06671 

14.9898 

BO 

.01455 

68.7501 

.03201 

31.2416 

.04949 

20.2056 

.06700 

14.9244 

51 

.01484 

67.4019 

.03230 

30.9599 

.04978 

20.0872 

.06730 

14.8596 

52 

.01513 

66.1055 

.03259 

30.6833 

.05007 

19.9702 

.06759 

14.7954 

531   .01542 

64.8580 

.03288 

30.4116 

.05037 

19.8546 

.06788 

14.7317 

54 

.01571 

63.6567 

.03317 

30.1446 

.05066 

19.7403 

.06817 

14.6685 

55 

.0160C 

(52.4992 

.03346 

29.8823 

.05095 

19.6273 

.06847 

14.6059 

56 

.01629 

61.3829 

.03376 

29.6245 

.05124 

19.5156 

.06876 

14.5438 

57!   .01658 

60.3058 

.03405 

29.3711 

.05153 

19.4051 

.06905 

14.4823 

58 

.01687 

59.2659 

.03434 

29.1220  ; 

.05182 

19.2959 

.06934 

14.4212 

r.i» 

.01716 

58.2612 

.03463 

28.8771  j 

.05212 

19.1879 

.06963 

14.3607 

60 

.01746 

57.2900 

.0341)2 

28.6363 

.05241 

19.0811 

.06993 

14.3007 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang  i   Tang 

89° 

88°                           87<> 

86° 

348 


TABLE   XXI.-TANfiF.NT3   AND   COTANGENTS. 


Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

' 

Ti 

.06993 

14.3007 

.08749 

11.4301 

.10510 

9.51436 

.12278 

8.14435 

60 

i 

.07023 

14.2411 

.08778 

11.3919 

.10540 

9.48781 

.12308 

8.12481 

59 

2 

.07051 

14.1821 

.08807 

11.3540 

.10569 

9.46141 

.123-J8 

8.10536 

r,s 

8 

.07U80 

14.  12;« 

.08837 

11.3163 

.10599 

9.413515  ! 

.12367 

8.08600 

57 

4 

.07110 

14.0655 

.08866 

11.2789 

.10628 

9.40904  i 

.12397 

8.06674 

56 

6 

.07139 

14.0079 

.08895 

11.2417 

.10657 

9.38307 

.12426 

8.04756 

55 

0 

.07168 

13.9507 

.08925 

11.2048 

i   .10687 

9.35724 

.12456 

8.02848 

54 

7 

.07197 

13.8940  ' 

.08954 

11.1081 

.10710 

9.33155 

.12485 

8.00948 

58 

s 

.07227 

13.8378 

.08983 

11.1316 

!    .10746 

9.30599  i 

.12515 

7.99058 

59 

9 

.07256 

13.7821 

.09013 

11.0954 

.  10775 

9.28058  i 

.12544 

7.97176 

51 

10 

.07285 

13.7267 

.09042 

11.0594 

|   .10805 

9.25530 

.12574 

.95302 

50 

11 

.07314 

13.6719 

.09071 

11.0237 

.10834 

9.23016  i 

.12603 

.93438 

4'.) 

18 

.07344 

13.6174 

.09101 

10.9882 

.10863 

9.20516 

.12633 

.91582 

48 

13 

.07373 

13.5634 

.09130 

10.9529 

.10893 

9.18028 

.12662 

.89734 

47 

11 

.07402 

13.5098 

.09159 

10.9178 

.10922 

9.15554 

.12692 

.87895 

413 

ir> 

.07431 

13.4566 

.09189 

10.8829 

|   .10952 

9.13093 

.12722 

.86064 

4~> 

16 

.07461 

13.4039 

.09218 

10.8483 

.10981 

9.10646 

.12751 

.84242 

44 

17 

.07490 

13.3515 

.09247 

10.8139 

.11011 

9.08211 

.12781 

.82428 

48 

i;, 

.07519 

13.2996 

.09277 

10.7797 

.11040 

9.05789 

.12810 

.80622 

48 

l:i 

.07548 

13.2480 

.09306 

10.7457 

.11070 

9.03379 

.12840 

.78825 

11 

90 

.07578 

13.1969 

.09335 

10.7119 

1   .11099 

9.00983 

.12869 

.77035 

to 

21 

.07607 

13.1461 

.09365 

10.6783 

.11128 

8.98598 

.12899 

.75254 

39 

x^ 

.07636 

13.0958 

.09394 

10.6450 

.11158 

8.96227 

.12929 

.73480 

3ti 

28 

.07665 

13.0458 

.09423 

10.6118 

.11187 

8.93867 

.12958 

.71715 

87 

-i 

.07695 

12.9962 

.09453 

10.5789 

.11217 

8.91520 

.12988 

.69957 

36 

25 

.07724 

12.9469 

.09482 

10.5462 

.11246 

8.89185 

.13017 

.68208 

85 

26 

.07753 

12.8981 

.09511 

10.5136' 

.11276 

8.86862 

.13047 

.66466 

84 

27 

.07782 

12.8496 

.09541 

10.4813 

.11305 

8.84551 

.13076 

.64732 

88 

28 

.07812 

12.8014 

.09570 

10.4491 

!    .11335 

8.82252 

.13106 

.63005 

38 

29 

.07041 

12.7536 

.09600 

10.4172 

.11364 

8.79964 

.13136 

.61287 

81 

30 

.07870 

12.7062 

.09629 

10.3854 

j    .11394 

8.77689 

.13165 

.59575 

30 

31 

.07899 

12.6591 

.09658 

10.3538 

!   .11423 

8.75425 

.13195 

.57872 

eo 

32 

.07'929 

12.6124 

.09688 

10.3224 

!   .11452 

8.73172 

.13224 

.56176 

28 

88 

.07958 

12.5660 

.09717 

10.2913 

.11482 

8.70931 

.13254 

.54487 

87 

34 

.07987 

12.5199 

.09746 

10.2602 

!   .11511 

8.68701 

.13284 

.52806 

26 

85 

.08017 

12.4742 

.09776 

10.2294 

.11541 

8.66482 

.13313 

.51132 

25 

38 

.08046 

12.4288 

.09805 

10.1988 

.11570 

8.64275 

.13343 

.49465 

24 

87 

.08075 

12.3838 

.09834 

10.1683 

|   .11600 

8.62078 

.13372 

.47806 

23 

88 

.03104 

12.3390 

.09864 

10.1381 

1    .11629 

8.59893 

.13402 

.46154 

22 

:.!) 

.08134 

12.2946 

.09893 

10.1080 

.11659 

8.57718 

.1:3432 

.44509 

21 

40 

.08163 

12.2505 

.09923 

10.0780 

.11688 

8.55555 

.13461 

.42871 

20 

41 

.08192 

12.2067 

.09952 

10.0483 

.11718 

F.  63402 

.13491 

.41240 

10 

.;:j 

.08221 

12.1632 

.09981 

10.0187 

i    .11747 

8.51269 

.13521 

.39616 

IS 

43 

.08251 

12.1201 

.10011 

9.98931 

i   .11777 

8.49128 

.13550 

.37999 

17 

44 

.08280 

12.0772 

.10040 

9.96007 

.11806 

8.47007 

.13580 

.36389 

16 

45 

.08309 

12.0346 

.10069 

9.93101 

.11836 

8.44896 

.13609 

.34786 

15 

46 

.08339 

11.9923 

.10099 

9.90211 

.11865 

8.42795 

.13639 

.33190 

14 

..'- 

.08368 

11.9504 

.10128 

9.87338 

.11895 

8.40705 

.13669 

.31600 

18 

« 

.08397 

11.9087 

.10158 

9.84482 

.11924 

8.38625 

.13698 

.30018 

12 

49 

.08427 

11.8673 

.10187 

9.81641 

.11954 

8.36555 

.13728 

.28442 

11 

50 

.08456 

11.8262 

.10216 

9.T8817 

.11983 

8.34496 

.13758 

.26873 

10 

51 

.08485 

11.7853 

.10246 

9.76009 

.12013 

8.32446 

.13787 

.25310 

'.• 

53 

.08514 

11.7448 

.10275 

9.73217 

.12042 

8.30406 

.13817 

.23754 

8 

58 

.Oa544 

11.7045 

.10305 

9.70441 

.12072 

8.28376 

.13846 

.22204 

7 

64 

.08573 

11.6645 

.10334 

9.67680 

.12101 

8.26355 

.13876 

.20661 

8 

65 

.08602 

11.6248 

.10363 

9.64935 

.12131 

8.24345 

.13906 

.19125 

6 

66 

.08632 

11.5853 

.10393 

9.62205 

.12160 

8.22344 

.13935 

.17594 

4 

67 

.086(51 

11.5461 

.10422 

9.59490 

.12190 

8.20352 

.13965 

.16071 

3 

68 

.08690 

11  5072 

.10452 

9.56791 

.12219 

8.18370 

.13995 

.14553 

2 

59 

.08720 

11.4685 

•   .10481 

9.54106 

.12249 

8.16398 

.14024 

.13042 

1 

60 

.08749 

11.4301 

.  10510 

9.51436 

.12278 

8.14435 

.14054 

7.11537 

0 

f 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

8 

5°           I 

8 

4° 

8 

3° 

8 

2° 

TABLE  XXI.— TANGENTS  AND  COTANGENTS. 


8°            i           9° 

10°            !,            IP 

Tang  |  Cotang  |j  Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0    .14054 

7.11537 

.15838 

6.31375 

.17633 

5.67128 

.19438 

5.14455    60 

1 

.14084 

7.10038 

.15868 

6.30189 

.17663 

5.66165 

.19468 

5.13658    59 

2 

.11113 

7.08.546 

.15898 

6.29007 

.17693 

5.65205 

.19498 

5.12862    58 

3;   .14143 

7.07059 

.15928 

6.27829 

.17723 

5.64248 

.19529 

5.12069    57 

4    .14173 

7.05579 

.15958 

6.26655 

.17753 

5.6321)5 

.19559 

5.11279    56 

5 

.14202 

7.04105 

.15988 

6.25186 

.17783 

5.62344 

.19589 

5.10490    55 

(j 

.14232 

7.02637 

.16017 

6.24321 

.17813 

5.61397 

.19619 

5.09704    54 

.14262  1  6.91174 

.16047 

6.23160 

.17843 

5.60452 

.19649 

5.08921    53 

g 

.14291  i  6.99718 

.16077 

6.22003 

.17873 

5.59511 

.19680 

5.08139    52 

8 

.14321 

6.98268 

.16107 

6.20851 

.17903 

5.58573 

.19710 

5.07360    51 

10 

.14351 

6.96823 

.16137 

6.19703 

.17933 

5.57638 

.19740 

5.06584    50 

11 

.14381 

G.  95385 

.16167 

6.18559 

.17963 

5.56706 

.39770 

5.05809    49 

12 

.14410 

6.93952 

.16196 

6.17419 

.17993 

5.55777 

.19801 

5.05037    48 

13 

.14440 

6.92525 

.16226 

6.16283 

.18023 

5.54851 

.19831 

5.  042:,  7    47 

14 

.14470 

6.91104 

.16256 

6.15151 

.18053 

5.53927 

.19861 

5.03499    46 

15     .14199 

6.89688 

.16286 

6.14023 

.18083 

5.53007 

.19891 

5.02734    45 

16 

.14529 

6.88278 

.16316 

6.12899 

.18113 

5.52090 

.19921 

5.01971    44 

17 

.14559     6.86874 

.16346 

6.11779 

.18143 

5.51176 

.19952 

5.01210    43 

18 

.14588     6.85475 

.16376 

6.10664 

.18173 

5.50264 

.10982 

5.00451    42 

19 

.14618 

6.840S2 

.16405 

6.09552 

.18203 

5.49356 

.20012 

4.99695    41 

90 

.14648 

6.82694 

.16435 

6.08444 

.18233 

5.48451 

.20042 

4.98940    40 

21 

.  14678 

6.81312 

.16465 

6.07340 

.18263 

5.47548 

.20073 

4.98188    39 

i>  > 

.14707 

6.79936 

.16495 

6.06240 

.18293 

5.46648 

.20103 

4.97438    38 

23     .14737 

6.78564 

.16525 

6.05143 

.18323 

5.45751 

.20133 

4.96690    37 

24  |   .14767 

6.77199 

.16555 

6.04051 

.18353 

5.44857 

.20164 

4.95945    36 

25 

.14796 

6.75838  ' 

.16585 

6.02962 

.18384 

5.43966 

.20194 

4.95201    35 

96 

.14826 

6.74483  1 

.16615 

6.01878 

.18414 

5.43077 

.20224 

4.94460    34 

27 

.14856 

6.73133 

.16645 

6.0079? 

.18444 

5.42192 

.20254 

4.93721    33 

28  i   .14886 

6.71789 

.16674 

5.99720 

.18474 

5.41309 

.20285 

4.92984    & 

29 

.11915 

6.70450 

.16704 

5.98646 

.18504 

5.40429 

.20315 

4.92249    31 

30 

.14945 

6.69116 

.16734 

5.97576 

.18534 

5.39552 

.20345 

4.91516    30 

31 

.14975 

6.67787 

.16764 

5.96510 

.18564 

5.38677 

.20376 

4.90785    29 

3;.' 

.15005 

6.66463 

.16794 

5.95448 

.18594 

5.37805 

.20406 

4.90056    28 

83 

.15034 

6.65144 

.16824 

5.94390 

.18624 

5.36936 

.20436 

4.89330    27 

3J 

.15064 

6.63831 

.16854 

5.93335 

.18654 

5.36070 

.20466 

4.88605    26 

35 

.15094 

6.62523 

.16884 

5.92283 

.18684 

5.35206 

.20497 

4.87882    25 

86 

.15124 

6.61219 

.16914 

5.91236 

.18714 

5.34345 

.20527 

4.87162    24 

J7 

.15153 

6.59921 

.16944 

5.90191 

.18745 

5.33487 

.20557 

4.86444    23 

38 

.15183 

6.5S627  ' 

.16974 

5.89151 

.18775 

5.32631 

.20588 

4.85727    22 

3D 

.15213 

6.57339 

.17004 

5.88114 

.18805 

5.31778 

.20618 

4.85013    21 

40 

.15243 

6.56055 

.17033 

5.87080 

.18835 

5.30928 

.20648 

4.84300    20 

41 

.15272 

C.  54777 

.17063 

5  86051 

.18865 

5.30080 

.20679 

4.83590    19 

4;! 

.15302 

6.53503 

.17093 

5.85024 

.18895 

5.29235 

.20709 

4.82882    18 

43 

.15332 

6.52234 

.17123 

5.840C1 

.18925 

5.28393 

.20739 

4.8217'5    17 

44 

.15362 

6.50970 

.17153 

5.82982 

.18955 

5.27553 

.20770 

4.81171    16 

45 

.15391 

6.49710 

.17183 

5.81966 

.18986 

5.26715 

.20800 

4.807'69    15 

46 

.  15421 

6.48456 

.17213 

5.80953 

.19016 

5.25880 

.20830 

4.80068    14 

47 

.15451 

6.47206 

.17243 

5.79944 

.19046 

5.25048 

.20861 

4.79370    13 

48 

.15481 

6.45961 

.17273 

5.78938 

.1907'6 

5.24218 

.20891 

4.78673    12 

19 

.15511 

6,44720 

.17303 

5.77936 

.19106 

5.23391 

.20921 

4.77978    11 

50 

.15640 

6.43484 

.17333 

5.76937 

.19136 

5.22566 

.20952 

4.77286    I" 

51 

.15570 

6.42253 

.17363 

5.75941 

.19166 

5.21744 

.20982 

4.76595      9 

52 

.15600 

6.41026 

.17393 

5.74949 

.19197 

5.20925 

.21013 

4.7590G      8 

.',-.! 

.15630 

6.39804 

.17423 

5.73960 

.1922? 

5.20107 

.21043 

4.75219      7 

51 

.15660 

6.38587 

.17453 

5.72974 

.19257 

5.19293 

.21073 

4.74534      6 

f:  :':> 

.15689 

6.37374 

.17483 

5.71992 

.19287 

5.18480 

.21104 

4.73851      5 

56 

.15719 

6.36165 

.17513 

5.71013 

.19317 

5.17671 

.21134 

4.73170      4 

57 

.15749 

6.34961 

.17543 

5.70037  i 

.19347 

5.16863 

.21164 

4.72190      3 

58 

.15779 

6.33761 

.17573 

5.69064 

.19378 

5.16058 

.21195 

4.71813     2 

59 

.15809 

6.32566 

.17603 

5.68094  ' 

.19408 

5.15256 

.21225 

4.71137      1 

;o 

.15838 

6.31375 

.17633 

5.67128 

.19438 

5.14455 

.21256 

4.70463      0 

/ 

Cotang 

Tang 

Cotang 

Tang      Cotang  j    Tang 

Cotang 

Tang 

81°           I1           80°           il           79°         '  '            78° 

TABLE   XXI.— TANGENTS   AND   COTANGENTS. 


4° 

1            5° 

6°                              7° 

Tang     Cotang 

Tang 

Cotang 

Tang     Cotang   i   Tang     Cotang 

0 

.06993 

14.3007 

.08749 

11.4301 

.10510     9.51436 

.12278 

8.14435  :60 

1     .07022 

14.2411 

.08778 

11.3919 

!   .10540 

9.48781  ; 

.12308 

8.12481 

si 

2     .07051 

14.1821 

.08807 

11.3540 

.10569 

9.46141 

.Itfi'JS 

8.10530 

56 

3;   .07080 

14.1285 

.08837 

11.3163 

.10599 

9.43515 

.12367 

8.08600 

57 

4     .07110 

14.0655 

.08866 

11.2789 

.10028 

9.40904  ;     .12397 

8.06674 

56 

5 

.07139 

14.0079 

.08895 

11.2417 

.10657 

9.38307 

.12426 

8.04756 

55 

G 

.07168 

13.9507 

.08925 

11.2048 

1   .10687 

9.35724 

.12456 

8.02848 

54 

7 

.07197 

13.8940 

.08954 

11.1081 

.10710 

9.33155 

.12485 

8.00948 

58 

8     .07237 

13.8378 

.08983 

11.1316 

i   .10746 

9.30599 

.12515 

7.99058 

58 

9     .07250 

13.7821 

.09013 

11.0954 

.  1077  5 

9.28058 

.12544 

7.97176 

51 

10 

.07285 

13.7267 

.09042 

11.0594 

.10805 

9.25530 

.12574 

7.95302 

50 

11 

.07314 

13.6719 

.09071 

11.0237 

1   .10834 

9.23016 

.12603 

7.93438 

49 

121   .07344 

13.6174 

.09101 

10.9882 

i   .10863 

9.20516 

.12633 

7.91582 

48 

13 

.07373 

13.5634 

.09130 

10.9529  i!    .10893 

9.18028 

.12662 

7.89734 

47 

14 

.07402 

13.5098 

:   .09159 

10.9178    1   .10922 

9.15554 

.12692 

7.87895    46 

15     .07431 

13.4566 

.09189 

10.8829 

.10952 

9.13093 

.12728 

7.86064    45 

10 

.07461 

13.4039 

.09218 

10.8483 

.10981 

9.10646 

.12751 

7.84242  ?44 

17 

.07490 

13.3515 

.09247 

10.8139 

.11011 

9.08211 

.12781 

7.82428  1  43 

18 

.07519 

13.2996 

1   .09277 

10.7797 

.11040 

9.05789 

.12810 

7.80622 

48 

19  j   .07548 

13.2480 

.09306 

10.7457  I     .11070 

9.03379 

.12840 

7.78825 

41 

20 

.07578 

13.1969 

.09335 

10.7119  II   .11099 

9.00983 

.12869 

7.77035 

10 

21 

.07607 

13.1461 

.09365 

10.6783 

.11128 

8.98598 

.12899 

7.75254 

39 

22     .07636 

13.0958 

.09394 

10.6450 

.11158 

8.96227 

.12929 

7.73480 

38 

23  |   .07665 

13.0458 

.09423 

10.6118 

.11187 

8.93867 

.12958 

7.71715 

87 

24 

.07085 

12.9962 

.09453 

10.5789 

.11217 

8.91520 

.12988 

7.69957 

36 

25 

.07724 

12.9469 

.09482 

10.5462 

.11246 

8.89185 

.13017 

7.68208 

35 

26 

.07753 

12.8981 

.09511 

10.5136' 

.11276 

8.86862 

.13047 

7.66466 

34 

27 

.07782 

12.8496 

.09541 

10.4813 

.11305 

8.84551 

.13076 

7.64732 

38 

2s 

.07812 

12.8014 

.09570 

10.4491 

.11335 

8.82252 

.13106 

7.63005 

32 

29 

.07841 

12.7536 

.09600 

10.4172 

.11364 

8.79964 

.13136 

7.61287 

81 

80 

.07870 

12.7062 

.09629 

10.3854 

.11394 

8.77689 

.13165 

7.59575 

30 

81 

.07899 

12.6591 

.09658 

10.3538 

.11423 

8.75425 

.13195 

7.57872 

& 

32 

.07929 

12.6124 

.09688 

10.3224    i    .11452 

8.73172 

.13224 

7.56176 

28 

88 

.07958 

12.5660 

.09717 

10.2913       .11482 

8.70931 

.13254 

7.54487    27 

34 

.07987 

12.5199 

.09746 

10.2602       .11511 

8.68701 

.13284 

7.52806    26 

85 

.08017 

12.4742 

.09776 

10.2294       .11541 

8.66482 

.13313 

7.51132 

•25 

36 

.08046 

12.4288 

.09805 

10.1988  I     .11570 

8.64275 

.13343 

7.49465    24 

37 

!  68075 

12.3838 

.09834 

10.1683 

!   .11600 

8.62078 

.13372 

7.47806    23 

38 

.03104 

12.3390 

.09864 

10.1381 

.11629 

8.59893 

.13402 

7.46154    22 

89 

.08134 

12.2946 

.09893 

10.1080 

.11659 

8.57718 

.13432 

7.44509 

•21 

40 

.08163 

12.2505 

.09923 

10.0780 

.11688 

8.55555 

.13461 

7.42871 

•20 

41 

.08192 

12.2067 

.09952 

10.0483 

.11718 

F.  63402 

.13491 

7.41240 

19 

42 

.08221 

12.1632 

.09981 

10.0187 

.11747 

8.51259 

.13521 

7.39616 

18 

48 

.08251 

12.1201 

.10011 

9.98931 

.11777 

8.49128 

.13550 

7.37999 

IT 

-1! 

.08280 

12.0772 

.10040 

9.96007 

.11806 

8.47007 

.13580 

7.36389    16 

46 

.08309 

12.0346 

.10069 

D.  93101 

.11836 

8.44896 

.13609 

7.34786    15 

46 

.08339 

11.9923 

.10099 

9.90211 

.11865 

8.42795 

.13639 

7.33190 

14 

,-- 

.08368 

11.9504 

.10128 

9.87338 

.11895 

8.40705 

.13669 

7.31600 

18 

£ 

.08397 

11.9087 

.10158 

9.84482 

.11924 

8.38625 

.13698 

7.30018 

12 

49 

.08427 

11.8673 

.10187 

9.81641 

.11954 

8.36555 

.13728 

7.28442 

11 

60 

.08456 

11.8262 

.10216 

9.78817 

.11983 

8.34496 

.13758 

7.26873 

10 

51 

.08485 

11.7853 

.10246 

9.76009 

.12013 

8.32446 

.13787 

7.25310 

9 

58 

.08514 

11.7448 

.10275 

9.73217 

.12042 

8.30406 

.13817 

7.23754 

8 

53 

.0&544 

11.7045 

.10305 

9.70441 

.12072 

8.28376 

.13846 

7.22204 

7 

54 

.OS573 

11.6645 

.10334 

9.67680 

.12101 

8.26355 

.13876 

7.20661 

6 

56 

.08602 

11.62-18 

.10363 

9.64935 

.12131 

8.24345 

.13906 

7.19125 

5 

58 

.08632 

11.5853 

.10393 

9.62205 

.12160 

8.22344 

.13935 

7.17594 

4 

57 

.08681 

11.5461 

.10422 

9.59490 

.12190 

8.20352 

.13965 

7.16071 

3 

58 

.08690 

11  5072 

i    .10452 

9.56791 

.12219 

8.18370 

.13995 

7.14553 

2 

59 

.08720 

11.4685 

.10481 

9.54106 

.12249 

8.16398 

.14024 

7.13042 

1 

60 

.08749 

11.4301 

.10510 

9.51436 

.12278 

8.14435 

!    .14054 

7.11537 

0 

/ 

Cotang     Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

85° 

i           84° 

83° 

82° 

TABLE  XXI— TANGENTS  AND  COTANGENTS. 


I,              8° 

9° 

10°                       11° 

Tang  |  Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

~0     .14054      7.11537 

.15838 

G.  31375 

.17633 

5.67128 

.19438 

5.14455 

1 

.14084 

7.10038 

.15868 

6.30189 

.17663 

5.66165 

.19468 

5.13658 

2 

.11113 

7.08546 

.15898 

6.29007 

.17693 

5.65205 

.19498 

5.12862 

3     .14143 

7.07059 

.15928 

6.27829 

.17723 

5.64248 

.19529 

5.12069 

41   .14173 

7.05579 

.15958 

6.26655 

.17753 

5.63295 

.19559 

5.11279 

5 

.  14302 

7.04105 

.15988 

6.25486 

.17783 

5.62344 

.19589 

5.10490 

6 

.14232 

7.02637 

.16017 

6.24321 

.17813 

5.61397 

.19619 

5.09704 

3 

.14262 

6.91174 

.16047 

6.23160 

.17843 

5.60452 

.19649 

5.08921 

8 

.14291      6.99718 

.16077 

6.22003 

.17873 

5.59511 

.19680 

5.08139 

9 

.14321 

6.98268 

.16107 

6.20851 

.17903 

5.58573 

.19710 

5.07360 

JO 

.14351 

6.96823 

.16137 

6.19703 

.17933 

5.57638 

.19740 

5.0G584 

11 

.14381 

G.  95385 

.16167 

6.18559 

.17963 

5  .  56706 

.39770 

5.05809 

12 

.14410 

6.93952 

.16196 

6.17419 

.17993 

5.55777 

.19801 

5.05037 

13 

.14440 

6.92525 

.16226 

6.16283 

.18023 

5.54851 

.19831 

5.0421.7 

11 

.14470 

6.91104 

.16256 

G.  15151 

.18053 

5.53927 

.19861 

5.03499 

15 

.14199 

G.  89688 

.16286 

6.14023 

.18033 

5.53007 

.19891 

5.02734 

16 

.14529 

6.88278 

.16316 

6.12899 

.18113 

5.52090 

.19921 

5.01971 

17 

.14559 

6.86874 

.16346 

6.11779 

.18143 

5.51176 

.19952 

5.01210 

18 

.14588 

G.  85475 

.16376 

6.10664 

.18173 

5.50264 

.1^982 

5.00451 

19 

.14618 

6.84082 

.16405 

6.09552 

.18203 

5.49356 

.20012 

4.99695 

20 

.14648 

6.82694  1 

.16435 

6.08444  I 

.18233 

5.48451 

.20042 

4.98940 

21 

.  14G78 

6.81312 

.16465 

6.07340 

.18263 

5.47548 

.20073 

4.98188 

22 

.  14707 

6.79936 

.16495 

6.06240 

.18293 

5.46648 

.20103 

4.97438 

2;J> 

.14737 

6.?'8564 

.16525 

6.05143 

.18323 

5.45751 

.20133 

4.96690 

24 

.14767 

6.77199 

.16555 

6.04051 

.18353 

5.44857 

.20164 

4.95945 

25 

.14796 

6.75838  ' 

.16585 

6.02962 

.18384 

5.43966 

.20194 

4.95201 

26 

.14826 

6.74483 

.16615 

6.01878 

.18414 

5.43077 

.20224 

4.94460 

27 

.14856 

6.73133 

.16645 

6.0079^ 

.18444 

5.42192 

.20254 

4.93721 

2S 

.14886 

6.71789 

.16674 

5.99720 

.18474 

5.41309 

.20285 

4.92984 

29 

.14915 

6.70450 

.16704 

5.98646 

.18504 

5.40429 

.20315 

4.92249 

30 

.  14945 

6.69116 

.16734 

5.97576 

.18534 

5.39552 

.20345 

4.91516 

81 

.14975 

6.67787 

.16764 

5.96510 

.18564 

5.38677 

.20376 

4.90785 

32 

.  15005 

6.66463 

.16794 

5.95448 

.18594 

5.37805 

.20406 

4.90056 

33 

.15034 

6.65144 

.16824 

5.94390 

.18624 

5.36936 

.20436 

4.89330 

34 

.15064 

6.63831 

.16854 

5.93335 

.18654 

5.36070 

.20466 

4.88605 

35 

.15094 

6.62523 

.16884 

5.92283 

.18684 

5.35206 

.20497 

4.87882 

36 

.15124 

6.61219 

.16914 

5.91236 

.18714 

5.34345 

.20527 

4.87162 

87 

.15153 

6.59921 

.16944 

5.90191 

.18745 

5.33487 

.20557 

4.86444 

38 

.15183 

6.58627  ' 

.16974 

5.89151 

.18775 

5.32631 

.20588 

4.85727 

89 

.15213 

6.57339 

.17004 

5.88114 

.18805 

5.31778 

.20618 

4.85013 

40 

.15243 

6.56055 

.17033 

5.87080 

.18835 

5.30928 

.20648 

4.84300 

41 

.15272 

6.54777 

.17063 

5  86051 

.18865 

5.30080 

.20679 

4.83590 

48. 

.15302 

6.53503 

.17093 

5.85024 

.18895 

5.29235 

.20709 

4.82882 

43 

.15332 

6.52234 

.17123 

5.840C1 

.18925 

5.28393 

.20739 

4.8217'5 

44 

.15362 

6.50970 

.17153 

5.82982 

.18955 

5.27553 

.20770 

4.814T1 

45 

.15391 

6.49710 

.17183 

5.81966 

.18986 

5.26715 

.20800 

4.807'69 

46 

.15421 

6.48456 

.17213 

5.80953 

.19016 

5.25880 

.20830 

4.80068 

47 

.15451 

6.47206 

.17243 

5.79944 

.19046 

5.25048 

.20861 

4.79370 

48 

.15481 

6.45961 

.17273 

5.78938 

.19076 

5.24218 

.20891 

4.78673 

'i!< 

.15511 

6.44720 

.17303 

5.77936 

.19106 

6.23391 

.20921 

4.77978 

60 

.15540 

6.43484 

.17333 

5.76937 

.19136 

5.22566 

.20952 

4.77286 

M 

.15570 

6.42253 

.17363 

5.75941 

.19166 

5.21744 

.20982 

4.76595 

5-2 

.15600 

6.41026 

.17393 

5.74949 

.19197 

5.20925 

.21013 

4.75906 

63 

.15630 

6.39804 

.17423 

5.73960 

.19227 

5.20107 

.21043 

4.75219 

54 

.15660 

6.38587 

.17453 

5.72974 

.19257 

5.19293 

.21073 

4.74534 

55 

.15689 

6.37374 

.17483 

5.71992 

.192S7 

5.18480 

.21104 

4.73851 

56 

.15719 

6.36165 

.17513 

5.71013 

.19317 

5.17671 

.21134 

4.73170 

5? 

.15749 

6.34961 

.17543 

5.70037 

.19347 

5.16863 

.21164 

4.72190 

58 

.15779 

6.33761 

.17573 

5.69064 

.19378 

5.16058 

.21195 

4.71813 

59 

.15809 

6.32566 

.17603 

5.68094 

.19408 

5.15256 

.21225 

4.71137 

60 

.15838 

6.31375 

.17633 

5.67128 

.19438 

5.14455 

.21256 

4.70463 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang  j    Tang 

Cotang 

Tang 

81°           lj           80°           il           79° 

78° 

TABLE   XXL— TANliKNTS   AND  COTANUKNTS. 


12°           l|           13°           !            14°                       15° 

Tang 

Cotang 

Tang     Cotang 

Tang     Cotang 

Tang  |  Cotang 

f 

0 

.21256 

4.70463 

.23087 

4.33148 

.24933     4.01078 

.20795     3.73205 

60 

1 

.21286 

4.69791 

.23117 

4.32573 

.24904     4.00583 

.2(5820     3.72771 

59 

2 

.21316 

4.69121 

.23148 

4.32001 

.24995     4.00080  i 

.20857     3.72338 

58 

a 

.21347 

4.68452 

.23179 

4.31430 

.25026     3.995U2 

.20888     3.71907 

57 

4 

.21377 

4.67786 

.23209 

4.30860 

.25056     3.99099 

.26920     3.71476 

56 

5 

.21408 

4.67121 

.23240 

4.30291 

.25087 

3.98007 

.26951 

3.71046 

55 

6 

.21438 

4.66458 

.23271 

4.29724 

.25118 

3.98117 

.20982 

3.70016 

54 

7 

.21469 

4.65797 

.23301 

4.29159 

.25149 

3.97027 

.27013 

3.70188 

53 

8 

.21499 

4.65138 

.23332 

4.28595 

.25180 

3.97139 

.27044 

3.09761 

52 

9 

.215S9 

4.64480 

.23363 

4.28032 

.25211 

3.96051 

.27070 

3.69335 

51 

10 

.21560 

4.63825 

.23393 

4.27471 

.25242 

3.90105 

.27107 

3.68909 

50 

11 

.21590 

4.63171 

.23424 

4.26911 

.25273 

3.95080 

.27138 

3.68485 

49 

12 

.21621 

4.62518 

.23455 

4.26352 

.25304 

3.95196 

.27109 

3.68001 

48 

13 

.21651 

4.61868 

.23485 

4.25795 

.25335 

3.94713 

.27201 

3.07638 

47 

14 

.21682 

4.61219 

.23516 

4.25239 

.25366 

3.94232 

.27232 

3.67217 

46 

15 

.21712 

4.60572 

.23547 

4.24685 

.25397 

3.93751 

.27263 

3.00796 

45 

16 

.21743 

4.59927 

.23578 

4.24132 

.25128 

3.93271 

.27294 

3.60370 

44 

17 

.21773 

4.59283 

.23608 

4.23580 

.25459 

3.92793 

.27326 

3.65957 

43 

a8 

.21804 

4.58641 

.23639 

4.23030 

.25490 

3.92316 

.27357 

3.05538 

42 

19 

.21834 

4.58001 

.23670 

4.22481 

.25521 

3.91839 

.27388 

3.05121 

41 

90 

.21864 

4.57363 

.23700 

4.21933 

.25552 

3.91364 

.27419 

3.04705 

40 

21 

.21895 

4.56726 

.23731 

4.21387 

.25583 

3.90890 

.27451 

3.64289 

39 

83 

.21925 

4.56091 

.23762 

4.20842 

.25614 

3.90417 

.27482 

3.63874 

38 

83 

.21956 

4.55453 

.23793 

4.20298 

.25645 

3.89945 

.27513 

3.03401 

37 

21 

.21986 

4.54826 

.23823 

4.19756 

.25676 

3.8947'4 

.27545 

3.03048 

36 

25 

.J2017 

4.54196 

.23854 

4.19215 

.25707 

3.89004 

.27576 

3.62636 

35 

86 

.22047 

4.53568 

.23885 

4.18675 

.25738 

3.88536 

.2?'G07 

3.62224 

34 

27 

.22078 

4.52941 

.23916 

4.18137 

.25769 

3.88068 

.27038 

3.61814 

33 

88 

..22108 

4.52316 

.23946 

4.17600 

.25800 

3.87601 

.27070     3.61405 

32 

2:) 

.22139 

4.51693 

.23977 

4.17064 

.25831 

3.87136 

.277'01      3.60990 

31 

30 

.22109 

4.51071 

.24008 

4.16530 

.25862 

3.86671 

.27732 

3  60588    30 

31 

.22200 

4.50451 

.24039 

4.15997 

.25893 

3.86208 

.27764 

3.60181    29 

88 

.22231 

4.49832 

.24069 

4.15465 

.25924 

3.85745 

.27795 

3.59775 

28 

33 

.22261 

4.49215 

.24100 

4.14934 

.25955 

3.85284 

.27826 

3.59370 

27 

34 

.22292 

4.48600 

.24131 

4.14405 

.25986 

3.84824 

.27853 

3.58966 

26 

35 

.22322 

4.47986 

.24162 

4.13877 

.26017 

3.84364 

.27889 

3.58562 

25 

36 

.22353 

4.47374 

.24193 

4.13350 

.26048 

3.83906 

.27921 

3.58160 

24 

37 

.22383 

4.46764 

.24223 

4.12825 

26079 

3.83449 

.27952 

3.57758 

23 

38 

.22414 

4.46155 

.24254 

4.12301 

.26110 

3.82992 

.27983 

3.57357 

22 

39 

.22444 

4.45548 

.24285 

4.11778 

.26141 

3.82537 

.28015 

3.50957 

"1 

40 

.22475 

4.44942 

.24316 

4.11256 

.26172 

3.82083 

.28046 

3.56557 

20 

41 

.22505 

4.44338 

.24347 

4.10736 

.26203 

3.81630 

.28077 

3.56159 

19 

42 

.22536 

4.43735 

.24377 

4.10216 

.26235 

3.81177 

.28109 

3.55761 

18 

43 

.22567 

4.43134 

.24408 

4.09699 

.26266 

3.80726 

.28140 

3.55304 

17 

44 

.22597 

4.42534 

.24439 

4.09182 

,20297 

3.80276 

.28172 

3.54908 

16 

45    .22628 

4.41936 

.24470 

4.08666 

.26328 

3.79827 

.28203 

3.54573 

15 

46  1   .22658 

4.41340 

.24501 

4.08152 

.20359 

3.79378 

.28234 

3.54179 

14 

47  i   .22689 

4.40745 

.24532 

4.07639 

.28390 

3.78931 

.28260 

3.53785 

13 

48!   .22719 

4.40152 

.24562 

4.07127 

.26421 

3.78485 

.28297 

3.53393 

12 

49  |   .22750 

4.395GO 

.24593 

4.06616 

.20152 

3.78040 

.28329 

3.53001 

11 

50 

.22781 

4.38969 

.24624 

4.06107 

.20483 

3.77595 

.28360 

3.52609 

10 

51 

.22811 

4.38381 

.24655 

4.05599 

.26515 

3.77152 

.28391 

3.52219 

9 

58 

.22842 

4.37793 

.24686 

4.05092 

.20546 

3.76709 

.28423 

3.51829 

8 

53 

.22872 

4.37207 

.24717 

4.04586 

.26577 

3.76268 

.28454 

3.51441 

7 

51 

.22903 

4.3'J023 

.24747 

4.04081 

.26008 

3.75828 

.28486 

3.51053 

6 

55 

.22934 

4.3G040 

.24778 

4.03578 

.26039 

8.75388 

.28517 

3.50666 

5 

50 

.22964 

4.35459 

.24809 

4.03076 

.26670 

3.74950 

.28549 

3.50279 

4 

57 

.22995 

4.34879 

.24840 

4.02574 

.26701 

3.74512 

.28580 

3.49894 

3 

53 

.23026 

4.34300 

.24871 

4.02074 

.26733 

3.74075 

.28012 

3.49509 

2 

53 

.23056 

4.33723 

.24902 

4.01576 

.26764 

3.73640 

.28043 

3.49125 

1 

GO 

.23087 

4.33148 

.24933 

4.01078 

.26795 

3.73205 

.28675 

3.48741 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang  |    Tang 

/ 

77° 

76° 

75° 

74° 

TABLE  XXI.— TANGENTS  AND  COTANGENTS. 


1           16°           1           17° 

18° 

19° 

/ 

Tang     Cotang      Tang 

Cotang 

Tang   ;  Cotang 

Tang  j  Cotang 

~o 

.28675  !  3.48741 

.30573 

3.27085 

.32492 

3.07768 

.34433  1  2.90421    60 

i 

.28706  !  3.48:359 

.30605 

3.26745 

.32524 

3.07464 

.34465 

2.90147    59 

g 

.28738  1  3.47977 

.30637 

3.26406 

.32556 

3.07160 

.34498 

2.89873    58 

3 

.28769  •  3.47596 

.30669 

3.26067 

.32588 

3.06857 

.34530 

2.89600 

57 

4     "28800     a.ffllG 

.30700 

3.25729 

.32621 

3.06554 

.34563 

2.89327 

56 

5    .'28832     3.46837 

.30732 

3.25392 

.32653 

3.06252 

.34596 

2.89055 

55 

G     .28864     3.46458 

.30764 

3.25055 

.32685 

3.05950 

.34628 

2.88783 

54 

7    .28895 

3.46080 

.30796 

3.24719 

.32717 

3.05649 

.34661 

2.88511    53 

8     .28927 

3.45703 

.30828 

3.24383 

.32749 

3.05349 

.34693 

2.88240    52 

9     .28958 

3.43337 

.30860 

3.24049 

.32782 

3.05049 

.34726 

2.87970 

51 

10.   .28990 

3.44951 

.30891 

3.23714 

.32814 

3.04749 

.34758 

2  87700 

50 

11     .29021 

3.44576 

.30923 

3.23381 

.32846 

3.04450 

.34791 

2.87430 

49 

12  i   .29053 

3.44202 

.30955 

3.23048 

.32878 

3.04152 

.34824 

2.87161    48 

13     .29084 

3.43829 

.30987 

3.22715 

.32911 

3.03854 

.34856 

2.86892    47 

14     .29116 

3.43456 

.31019 

3.22384 

.-32943 

3.03556 

.34889 

2.86624 

46 

15     .211147 

3.43084 

.31051 

3.22053 

.32975 

3.03260 

.34922 

2.86356    45 

16     .29179 

3.42713 

.31083 

3.21722 

.33007 

3.02963 

.34954 

2.86089    44 

17     -29210 

3.42343 

.31115 

3.21392 

.33040 

3.02667 

.34987 

2.85822 

43 

H 

.29242 

3.41973  1 

.31147 

3.21063 

.33072 

3.02372 

.35020 

2.85555    42 

i:» 

.  29274 

3.41604 

.31178 

3.20734 

.33104 

3.02077 

.35052 

2.85289  1  41 

go 

.29305 

3.41236  | 

.31210 

3.20406 

.33136 

3.01783 

.35085 

2.85023 

40 

01 

.29337 

3.40869 

.31242 

3.20079 

.33169 

3.01489 

.35118 

S.  84758 

39 

22 

.29368 

3.40502 

.31274 

3.19752 

.33201 

3.01196 

.35150 

2.84494 

38 

28 

.29400 

3.40136 

.31306 

3.19426 

.33233 

3.00903 

.35183 

2.84229 

37 

-.'4 

.29432 

3.39771 

.31338 

3.19100 

.83360 

3.00611 

.35216 

2.83965  i36 

25 

.29463 

3.39406 

.31370 

3.18775 

.33298 

3.00319 

.35248 

2.83702    35 

26  i   .29495 

3.39012 

.31402 

3.18451 

.33330 

3.00028 

.35281 

2.83439 

34 

27  !   .29526 

3!  33079 

.31434 

3.18127 

.33363 

2.99738 

.35314 

2.83176 

33 

28     .29558 

3.38317 

.31466 

3.17804 

.33395 

2.99447 

.35346 

2.82914 

32 

29 

.29590 

3.37955 

.31498 

3.17481 

.33427 

2.99158 

.35379 

2.82653 

31 

30 

.29621 

3.37594 

.31530 

3.17159 

.33460 

2.98868 

.35412 

2.82391 

30 

31 

.29653 

3.37234 

.31562 

3.16838 

.33492 

2.98580 

.35445 

2.82130 

29 

32 

.29(585 

3.36875 

.31594 

3.16517 

.33524 

2.98292 

.35477 

2.81870 

28 

33    .20716 

3.3G516 

.31626 

3.16197 

.33557 

2.S8004 

.35510 

2.81610 

27 

34    .29748 

3.36158 

.31658 

3.15877 

.33589 

2.97717 

.35543 

2.81350 

26 

35 

.29780 

3.35800 

.31690 

3.15558 

.33621 

2.97430 

.35576 

2.81091 

25 

36    .29311 

3.35443 

.31722 

3.15240 

.33654 

2.97144 

.35608 

2.80833 

24 

37!   .29843 

3.3508? 

.31754 

3.14922 

.33686 

2.96858 

.35641 

2.80574 

23 

38      29'  '75 

3.34732 

,  .31786 

3.14605 

.33718 

2.96573 

.35674 

2.80316 

22 

39     .99906 

3.34377 

.31818 

3.14288 

.33751 

2.96288 

.35707 

2.80059 

21 

40    .29938 

3.34023 

.31850 

3.13972 

.33783 

2.96004 

.35740 

2.79802 

20 

41     .29970 

3.33670 

.31882 

3.13656 

.33816 

2.95721 

.35772 

2.79545 

19 

42    .30001 

3.33317 

.31914 

3.13341 

.33848 

2.95437 

.35805 

2.79289 

18 

43  j  .30033 

3.32965 

.31946 

3.13027 

.33881 

2.95155 

.35838 

2.79033 

17 

44 

.80065 

3.32814 

.31978 

3.12713 

.33913 

2.94872 

.35871 

2.78778 

16 

45 

.30097 

3.32264 

.32010 

3.12400 

.33945 

2.94591 

.35904 

2.78523 

15 

46    .30128 

3.31914 

.32042 

3.12087 

.33978 

2.94309 

.35937 

2.78269 

14 

47     .30160 

3.31565 

.32074 

3.11775 

.34010 

2.94028 

.35969 

2.78014 

13 

48     .30193 

3.31216 

.32106 

3.11464 

.34043 

2.93748 

.36002 

2.77761 

12 

49    .30224 

3.30S68 

.32139 

3.11153 

.34075 

2.93468 

.36035 

2.77507 

11 

50    .30255 

3.30521 

.32171 

3.10843 

.34108 

2.93189 

.36068 

2.77254 

10 

51     .30287 

3.30174 

.32203 

3.10532 

.34140 

2.92910 

.36101 

2.77002 

9 

52 

.30319 

3.29829 

.32235 

3.10223 

.34173 

2.92632 

.36134 

2.76750 

8 

53    .30351 

3.29483 

.32267 

3.09914 

-.34205 

2.92354 

.36167 

2.76498 

7 

54:   .30382 

3.29139 

.32299 

3.09606 

.34238 

2.92076 

.36199 

2.76247 

6 

55    .30414 

3.28795 

.32331 

3.09298 

.34270 

2.91799 

.36232 

2.75996 

5 

56 

.30446 

3.28452 

.32363 

3.08991 

.34303 

2.91523 

.36265 

2.75746 

4 

57 

.30478 

3.28109 

.32396 

3.08685 

.34335 

2.91246 

.36298 

2.75496 

3 

58 

.30509 

3.27767 

.32428 

3.08379 

.34368 

2.90971 

.36331 

2.75246 

2 

59 

.30541 

3.27426 

.32460 

3.0R073 

.34400 

2.90696 

.36364 

2.74997 

1 

60 

.30573     3.27085 

.32492 

3.07768 

.34433 

2.90421 

.36397 

2.74748 

0 

/ 

Cotanr' 

Tang 

Cotang 

Tang 

Cotang 

Tang      Cotang 

Tang 

/ 

V3°           il           72° 

71° 

70° 

352 


TABLE   XXT.-TANOENTS   AND   COTANGENTS. 


!           20°           ||          21° 

22°                       23° 

I  Tang  |  Cotuiig 

Tang 

Cotang 

Tang 

Cotang      Tang  |  Cotang 

0    .36397     2.74748 

!   .38386 

2.60509 

.  40403 

2.47509        .424ir" 

2.35585 

60 

1     .38430     2.74499 

.38420 

2.60283 

.40436 

2.47302       .42483 

2.35395 

59 

2    .36463 

2.74251 

.38453 

2.60057 

.40470 

a.  47095       .42516 

2.35205 

r,s 

3    .36496 

2.74004 

.38487 

2.59831 

.40504 

2.46888       .42551 

2.35015 

51! 

4    .86539 

2.73756 

.38520 

2.59G06 

.40538 

2.46682       .4&ZS5 

2.34825 

56 

5|   .36363 

2.73509 

.38553 

2.59381 

.40572 

2.46476       .42619 

2.34636 

55 

6    .36595 

2.73263 

.38587 

2.59156 

.40606 

2.46270 

.42054 

2.34447 

54 

7     .36623 

2.73017 

.38620 

2.58932 

.40640 

2.46065 

.43688 

2.34258 

53 

8    .36661 

2.72771 

.38654 

2.58708 

.40674 

2.45860 

.42722 

2.34069 

52 

9    .36694 

2.72526 

.38687 

2.58484 

.40707 

2.45655 

.42757 

2.33881 

5] 

10,   .36727 

2.72281 

.38721 

2.58261 

.40741 

2.45451 

.42791 

2.33693 

50 

11     .36760 

2.  72036 

.38754 

2.58038 

.40775 

2.45246 

.42826 

2.33505 

10 

12 

.36793 

2  71792 

.38787 

2.57815 

.40809 

2.45043 

.42800 

2.33317 

48 

13 

.36825 

Si  71548 

.38821 

2.57593 

.40843 

2.44839 

.42894 

g!  33130 

47 

1! 

.36859 

2.71305 

.38854 

2.57371 

.40877 

2.44636 

.42929 

2.82943 

46 

15 

.36893 

2.71083 

.38888 

2.57150 

.40911 

2.44433 

.42963 

2.32756 

K 

16 

.38925 

2.70819 

.38921 

2.56928 

.40945 

2.44230 

.42998 

2.32570 

H 

17 

.36953 

2.70577 

.38055 

2.56707 

.40979 

2.44027 

.43032 

2.32883 

43 

18    .36991 

2.70335 

.38988 

2.56487 

.41013 

2.43825 

.43067 

2.32197    42 

19    .37024 

2.70094 

.39022 

2.56266 

.41047 

2.43623 

.43101 

2.32013     11 

20     .37057 

2.69853 

.39055 

2.56046 

.41081 

2.4:3422 

.43136 

2.31826 

W 

211   .37090 

2.69612 

.39089 

2.55827 

.41115 

2.43220 

.43170 

2.31641 

39 

22     .37123 

2.69371 

.39122 

2.55608 

.41149 

2.43019 

.43205 

2.31456 

88 

23 

.37157 

2.69101 

.39156 

2.55389 

.41183 

2.42819 

.43230 

2.31271 

24 

.37190 

2.68892 

.39190 

2.5517'0 

.41217 

2.42618 

.43274 

2.31086 

86 

25    .37223 

2.68653 

.39223 

2.54952 

.41251 

2.42418 

.43308 

2.30902 

35 

26  1   .37256 

2.68414 

.39257 

2.54734 

.41285 

2.42218 

.43343 

2.30718 

84 

27 

.37289 

2.68175 

.39290 

2.54316 

.41319 

2.42019 

.43378 

2.80534 

83 

^ 

.37322 

2  67937 

.39324 

2.54299 

.41353 

2.41819 

.43412 

2.30351 

32 

29 

.37355 

2.67700 

!  39357 

2.54082 

.41387 

3.41620 

.43447 

2.30167    31 

80 

.37388 

2.67462 

.39391 

2.53865 

.41421 

2.41421 

.43481 

2.29984  J30 

31 

.37422 

2.67225 

.39425 

2.53648 

.41455 

2.41223 

.43516 

2.29801  !29 

82 

.37455 

2.66989 

.39453 

2.53432 

.41490 

2.41025 

.43550 

2.29619    28 

33 

.37483 

2.66752 

.39493 

2.53217 

.41524 

2.40827 

.43585 

2.29437    27 

34 

.37521 

2.66516 

.39528 

2.53001 

.41553 

2.40629 

.43620 

2.29254    26 

35 

.37554 

2.66281 

!   .39559 

2.52786 

.41592 

2.40432 

.43654 

2.29073    25 

86 

.37588 

2.66046 

.   .39593 

2.52571 

.41626 

2.40235 

.43689 

2.28891    21 

37 

.37621 

2.65811 

i   .39826 

2,52357 

.41660 

2.40038 

.43724 

2.28710    23 

88 

.3765i 

2.65576 

.39660 

2.52142 

.41694 

2.39841. 

.43758 

,'.'.2S52S    22 

39 

.37687 

2.65342 

.39894 

2.51929 

[41728 

2.39645 

.43793 

y.  28348    21 

•10 

.37720 

2.65109 

.39727 

2.51715 

.41763 

2.39449 

.43828 

2.28167    20 

41 

.37754 

2.64875 

.39761 

2.51502 

.41797 

2.39253 

.43862 

2.279S7    19 

42 

37787 

2.64643 

.39r93 

2.51289 

.41831 

2.39058 

.43897 

2.27806    18 

48 

.37823 

2.64410 

.39829 

2.51076 

.41865 

2.38863 

.43933 

2.27626    17 

44 

.37853 

2.64177 

.39362 

2.50864 

41899 

2.38668 

.43966 

2.27'447    16 

45 

37887 

2.63945 

.39896 

2.50653 

.41933 

2.38473 

.44001 

2.27267    15 

46 

.37920 

2.63714 

.39930 

2.50440 

.41968 

2  38279 

.44036 

2.27088  114 

47 

.37953 

2.63483 

.39963 

2.50229 

.42002 

2.'38084 

.44071 

2.26909    18 

18 

.37985 

2.63253 

.39997 

2.50018 

.42036 

2.37891 

.44105 

2.26730 

(2 

48 

.38020 

2.63021 

.40031 

2.49807 

.42070 

2.37697 

.44140 

2.26553 

11 

50 

.33053 

2.62791 

.40065 

2.49597 

.42105 

2.37504 

.44175 

2.26374 

10 

51 

.38088 

2.62561 

.40098 

2.49386 

.42139 

2.37311 

.44210 

2.26196 

a 

.38120 

2.62333 

.40132 

2.49177 

.42173 

2.37118 

.44244 

2.26018 

H 

53 

.38153 

2.62103 

.40166 

2.48967 

.42207 

2.3G925 

.44279 

2.25840 

7 

54 

.38186 

2.61874 

.40200 

2.48758 

.42242 

2.36733 

.44314 

2.25603 

(i 

55 

.38220 

2.61646 

.40234 

2.48549 

.42276 

2.36541 

.44349 

2.25486 

B 

56 

.38253 

2.61418 

.40267  I  2  48340 

.42310 

2.36349 

.44384 

2.25309 

4 

57 

.38286 

2.61190 

.40301     2.48132 

.42345 

2.36158 

.44418 

2.25132 

3 

58 

.38320 

2.60963 

.40335  !  2.47924 

.42379 

2.359fi7 

.44453 

2.24956 

% 

59 

!  38353 

2.(i()73ri 

.40369  !  2.47716 

.42413 

2.35776 

.44488     224780 

1 

60 

.38386 

2.60509 

.40403     2.47509  \     .42147 

2.35585 

.44523     2.24604      U 

/ 

Cotang  |    Tang 

Cotang  ;    Tang      Cotang     Tang 

Cotang     Tang 

> 

69°          !|           68*           II           67°          H          66° 

TABLE  xxi.— TAX<;F.XTS  AND  COTANGENTS. 


}  \           24°           '            25°            '           26°           i            27° 

Tang  i  Cotang   i  Tang 

Cotang      Tang     Cotang 

Tang     Cotang 

t. 

.44523 

2.24604 

.46631 

2.14451 

.48773 

2.05030 

"7511953"    1.96261    (50 

1 

.44558 

2.244.28 

.46666 

2.14288 

.48809 

2.04879 

.50989     1.96120  !59 

2 

.44593 

2.24252 

.46702 

2.14125 

.48845 

2.04728 

.51026 

1.95979    58 

3 

.44627 

2.24077 

.46737 

2.13963 

.48881 

2.04577 

.51063 

1.95838  !57 

•1 

.44662 

2.23903 

.46772 

2.13801 

.48917 

2.04426 

.51099 

1.95698  '56 

5 

.44697 

2.23727 

.46808 

2.13639 

.48953 

2.04276 

.51136 

1.95557    55 

G 

.44732 

2.23553 

.46843 

2.13477 

.48989 

2.04125 

.51173 

1.95417  ,54 

7 

.44767 

a!23378 

i   .46879 

2.13316 

.49026 

2.0397'5 

.51209 

1.95277    53 

6 

.44802 

2.2-5204 

.46914 

2.13154 

.49062 

2.03825 

.51240 

1.95137    52 

9 

.44837 

2.23030 

.46950 

2.12993 

.49098 

2.03S75 

.51283 

1.94997  151 

10 

.44873 

2.22857 

i    .46985 

2.12832 

.49134 

2.0352G 

.51319 

1.94858 

50 

1! 

.44907 

2.22683 

!    .47021 

2.12671 

.49170 

2.03376 

.51:356 

1.94718 

49 

12 

.44912 

2.22510 

.47056 

2.12511 

.49206 

2.03227 

.51393 

1.94579  '48 

13 

.44977     2.22337 

1    .47002 

2.12350 

.49242 

2.03078 

.51430 

1.94440 

47' 

14 

.45012 

2.22164 

.47128 

2.12190 

.49278 

2.02929 

.51467 

1.94301 

46 

15 

.45047 

2.21992 

.47163 

2.12030 

.49315 

2.02780 

.51503 

1.94162  '45 

in 

.45082     2.21819 

.47199 

2.11871 

.49351 

2.02631 

.51540 

1.94023    44 

ir 

.45117 

2.21647 

.47234 

2.11711 

.49387 

2.02483 

.51577 

1.93885    43 

is 

.45152 

2.21475 

.47270 

2.11552 

.49423 

2.02335 

.51614 

1.93746  !42 

19 

.45187 

2.21304 

.47305 

2.11392 

.49459 

2.02187 

.51G51 

1.93608    41 

90 

.45222 

2.21132 

.47341 

2.11233 

.49495 

2.02039 

.51688 

1.93470 

40 

81 

.45257 

2.20961 

.47377 

2.11075 

.49532 

2.01891 

.51724 

1.93332 

39 

22 

.45292 

2.20790 

.47412 

2.10916 

.49568 

2.01743 

.51761 

1.93195 

38 

23 

.45327 

2.20619 

.47443 

2.10758 

.49604 

2.01596 

.51798 

1.93057  |37 

24 

.45362 

2.20449 

.47483 

2.10600 

.49640 

2.01449  ! 

.51835 

1.92920  |36 

26 

.45397 

2.20378 

.47519 

2.10442 

.49677 

2.01302  i 

.51873 

1.92782 

35 

26 

.45432 

2.20108 

.47555 

2.10284 

.49713 

2.01155  1 

.51909 

1.92645 

34 

27 

.45467 

2.19938 

.47590 

2.10126 

.497'49 

2.01008 

.51946 

1.92508 

33 

28 

.45502 

2.19769 

.47626 

2.09969 

.497'86 

2.00862 

.51983 

1.92371    32 

29 

.45538 

2.19599 

.47662 

2.09811 

.49822 

2.00715 

.52020 

1.92235    31 

30 

.45573 

2.19430 

.47698 

2.09654 

.49858 

2.00569 

.52057 

1.92098 

30 

31 

.45608 

2.19261 

.47733 

2.09498 

.49894 

2.00423 

.52094 

i.  91  962 

29 

32 

.45643 

2.19092 

.47769 

2.09341 

.49931 

2.00277 

.52131 

1.91626 

28 

33 

.4567'8 

2.18923 

.47805 

2.09184 

.49967 

2.00131 

.521G8 

1.91G90 

^7 

84 

.45713 

2!  18755 

.47840 

2.09028 

.50004 

1.99986 

.52205 

1.91554 

26 

35 

.45748 

2.18587 

.47876 

2.08872 

.50040 

1.99841 

.52242 

1.91418 

25 

30 

.45784 

2.18419 

.47912 

2.08716 

.5007'6 

1.99695 

.52279 

1.91282 

24 

37 

.45819 

2.18251 

.47948 

3.08560 

.50113 

1.99550 

.52316 

1.91147 

23 

58 

.45854 

2.18084 

.47984 

2.08405 

•  .50149 

.99406 

.52353 

1.91012 

22 

31) 

.45889 

2.17916 

.48019 

2.08250 

.50185 

.99261 

.52390 

1.90876 

21 

40 

.45924 

2.17749 

.48055 

2.08094 

.50222 

.99116 

.52427 

1.90741 

20 

41 

.45960 

2.17582 

.48091 

2.07939 

.50258 

.98972 

.52464 

1.90607 

19 

42 

.45995 

2.17416 

.48127 

2.07785 

.50295  i     .98828 

.52501 

1.90472 

18 

43!   .46030 

2.17249 

.48163 

2.07630 

.50331  !     .98684  1 

.52538 

1.90337 

17 

44  j   .46065 

2.17083 

.48198 

2.07476 

.50368  |     .98540 

.52575 

1.90203 

16 

45 

.46101 

2.16917 

.48234 

2.07321 

.50404 

.98396 

.52613 

1.90069 

15 

46 

.46136 

2.16751 

.48270 

2.07167 

.50441 

.98253 

.52650 

1.89935  i!4 

47 

.46171 

2.16585 

.48306 

2.07014 

.50477 

.98110 

.52687 

1.8S801    13 

4S 

.46206 

2.16420 

.48342 

2.06860 

.50514 

1.97966 

.52724 

1.89667 

12 

4!) 

.46242 

2.1  6255 

.48378 

2.06706 

.50550 

1.97823 

.52761 

1.89533 

11 

50 

.46277 

2.16090 

.48414 

2.06553 

.50587 

1.97681 

.52798 

1.89400 

10 

51 

.46312 

2.15925 

.48450 

2.06400 

.50623 

1.97538 

.52836 

1.8926b 

9 

52 

.46348 

2.15760 

.48486     2.06247 

.50660 

1.97395 

.52873 

1.89133 

8 

53 

.46383 

2.15596 

.48521 

2.06094 

.50696 

1.97253 

.52910 

1.89000 

7 

54 

.46418 

2.15432 

.48557 

2.05942 

.50733 

1.97111 

.52947 

1.88867 

6 

55 

.46454 

2.15268 

.48593 

2.05790 

.50769 

1.96969 

.52985 

1.88734 

5 

56 

.46489 

2.15104 

.48629     2.05637 

.50806 

1.96827 

.53022 

1.88602     4 

57  1   .46525 

2.14940 

.48665     2.05485 

.50843 

1.96685 

.53059 

1.88469  1  3 

58  i   .46560 

2.14777 

.48701     2.05333 

.50879 

1.96544 

.53096 

1.88337 

2 

59     .46595 

2.14614 

.48737     2.05182 

.50916 

1.96402 

.53134 

1.88205 

1 

60 

.46631 

2.14451 

.48773     2.05030 

.50953 

1.96261 

.53171 

1.88073 

0 

J 

Cotang 

Taug      Cotang     Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

65° 

64° 

63° 

62° 

TABLE   XXI.— TAN(JKXTS    AND   r<  >TAN<  JKNTS. 


28°                       29°                       30° 

31° 

Tang 

Cotang 

Tang 

Cotang      Tang  |  Cotang 

Tang     Cotang 

Of   .53171 

1.88073 

.55431 

1.80405       .57735 

1.73205 

.ooost;     i.(;0428 

60 

j 

.53208 

1.87941        .55469 

1.80281       .57774 

1.73089 

.60126      1.66318 

59 

0 

.53246 

1.87809  .     .55507 

1.80158       .57813 

1.72973 

.60105      1.06209 

58 

8 

.53283 

1.87677       .55545 

1.80034       .57851 

1.7'2857  •     .60205  !  1.  (56099  |57 

4 

.53320 

1.87546 

.55583 

1.79911        .57890 

1.72741   j     .60245      1.65990  :Z6 

5 

.53358 

1.87415 

.55621 

1.79788  ,     .57929 

1.72625       .60284      1.65881  155 

(1 

.53395 

1.87283 

.55659 

1.79665       .57908 

1.72509       -6032J      1.6S772  T4 

7' 

.53432 

1.87152 

.55697 

1.79542       .58007 

1.72393  j!    .60364  ;  1.05663    5o 

8 

.53470 

1.87021 

.55736 

1.  71)41  9       .58046 

1.72278       .60403 

1.65554  JM 

0 

.53507 

1.86891 

.55774 

1.711296  !     .58085 

1.72163       .60443 

1.05445    51 

10 

.53545 

1.86760 

.55812 

1.79174 

.58124 

1.72047 

i    .00483 

1.05337 

BO 

11 

.53582 

1.86630 

.55850 

1.79051 

.58162 

1.71932    j   .60522 

1.65228 

49 

1-2 

.53620 

1.86499 

.55888 

1.  78929 

.58201 

1.71817  i!    .60562 

1.05100    4S 

13 

.53657 

1.86369 

.55926 

1.78807 

.58240 

1.71702       .(1000:2      1.05011   :47 

14    .53694 

1.86239    i   .55964 

1.78685 

.58279 

1.71588  i     .60642      1.645)03    40 

15    .53732 

1.86109    1   .56003 

1.78563 

.58318 

1.71473  i     .60681 

1.04795    45 

16    .53769 

1.85979 

.56041 

1.78441 

.58357 

1.71358  1     .60721 

1.040H7    44 

17     .53807 

1.85850 

.56079 

1.78319 

.58396 

1.71244 

.60761 

1.01571)    4:i 

18    .53844 

1.85720 

.56117 

1.78198 

.58435 

1.71129 

.60801 

1.64471     4.'2 

19:   .53882 

1.85591 

.56156 

1.78077 

.58474 

1.71015 

.OOS41 

1.C480S    41 

20   .:>:«fc>o 

1.85462 

.56194 

1.77955 

:   .58513 

1.70901 

.60881 

1.04250 

40 

21 

.53957 

1.85333 

.56232 

1.77834 

.58552 

1.70287 

1    .60921 

1.64148 

80 

22 

.53995 

1.85204 

.56270 

1.77713 

.58591 

1.70673 

.60960 

1.  MO  11    88 

28 

.54032 

1.85075 

.56309 

1.77592 

.58631 

1.70560 

:    .61000 

1.CS934    :j< 

24 

.54070 

1.84946 

.56347 

1.77471 

.58670 

1.70446 

.01040 

1.63820  ^30 

26 

.54107 

1.84818 

.56385 

1.77a51 

.58709 

1.70332 

.61080 

1.03719    85 

26 

.54145 

1.84689 

.56424 

1.77230 

!   .587'48 

1.70219 

.61120 

1.03612  |34 

27    .54183 

1.84561 

.56462 

1.77110 

.58787 

1.70106 

.61160 

1.03505 

83 

28 

.54220 

1.84433 

.56501 

1.76990 

.58826 

1.69992 

.01200 

1.63S98 

82 

29 

.54258 

1.84305 

.56539 

1.76869 

.58865 

1.69879 

.61240 

1.63292 

,-51 

80 

.54296 

1.84177 

.56577 

1.78749 

.58905 

1.69766 

.01280 

1.03185 

80 

31 

.54333 

1.84049 

.56616 

1.76629 

1   .58944 

1.69653 

1   .61320 

1.G3079 

89 

82 

.54371 

1.83922 

.50G54 

1.76510 

!   .58983 

1.C9541 

.61300 

1.0^972 

28 

S3 

.54409 

1.83794 

.56693 

1.76390 

.50022 

1.69428 

.61400 

1.62866 

27 

;n 

.54446 

1.83667 

.56731 

1.76271 

.59061 

1.69316 

.61440 

1.62700 

20 

35 

.54484 

1.83540 

.56769 

1.76151 

.59101 

1.69203 

.61480 

1.62654 

25 

36 

.54522 

1.83413 

.56808 

1.76032 

.59140 

1.69091 

.61520 

1.62548 

x>4 

37 

.54560 

1.83286 

.56846 

1.75913 

.59179 

1.68979 

.61561 

1.02442 

2;> 

38:   .54597 

1.83159 

.56885 

1.75794- 

1   .59218 

1.68866 

.61601 

1.02336 

22 

39 

.54635 

1.83033 

.56923 

1.75675 

.C9258 

1.68754 

.61641 

1.62230 

83 

40 

.54673 

1.82906 

.56962 

1.75556 

.59297 

1.68G43 

.61681 

1.62125 

2.) 

41 

.54711 

1.82780 

.57000 

1.75437 

.59336 

1.68681 

.61721 

1.02019 

19 

4.2 

.54748 

1.82654 

.57039 

1.75319 

.59376 

1.08419 

.61761 

1.01914 

18 

43 

.54786 

1.82528    1   .57078 

l!  75200 

1   .59415 

1.68308       .61801 

i  Gi  SOB 

17 

44 

.54824 

1.82402 

.57116 

1.75082 

.59454 

1.68196 

.61842 

1.01703 

10 

45 

.54862 

1.82276 

.57155 

1.74964 

.59494 

1.68085 

.61882 

1.01598 

15 

46 

.54900 

1.82150 

.57193 

1.74846 

i   .59533 

1.67974 

.61922 

1.01493 

14 

47 

.54938 

1.82025 

.57232 

1.74728 

.59573 

1.67863  :      61962 

1.61388 

18 

48 

.54975 

1.81899 

.57271 

1.74610 

.59612 

1  .  67752 

I   .62003 

1.01283 

12 

4!) 

.55013 

1.81774 

.57309 

1.74492 

.59651 

1.C7641 

.62043 

1.01179    11 

50 

.55051 

1.81649       .57348 

1.7437'5 

.59691 

1.67530 

.02083  j  1.01074 

10 

51 

.55089 

1.81524 

.57386 

1.74257 

.59730 

1.67419 

.62124  !  1.60970 

9 

52 

.55127 

1.81399 

.57425      1.74140 

.59770 

1.67309 

.62164      1.00865 

8 

53    .55165 

1.81274 

.57464 

1.74022 

:   .59809 

1.67198 

.62201      1.60761 

7 

54 

.55203 

1.81150 

.57503 

1.73905 

.59849 

1.67088 

.62245 

1.00057 

6 

55 

.55241 

1.81025 

.57541 

1.73788 

.59888 

1.66978 

.02285 

1.00553 

5 

66 

.55279 

1.80901 

.57580     1.73671 

.59928 

1.66867 

.02325      1.00449 

4 

57 

.55317 

1.80777 

.57619  !  1.73555 

.59967 

1.66757    :    .62800      1.00345 

3 

58!    .55355 

1.80653 

.57657      1.73438       .60007 

1.66647    i    .62406      1.00241 

2 

59j    .55393 

1.80529 

.57696      1.73321       .60046 

1.66538       .62446      1.60137 

i 

60    .55431 

1.80405       .57735      1.73205  |    .60086      1.06428 

.02487      1.00033 

0 

^  j  Cotang 

Tang      Cotang     Tang     !  Cotang     Tang 

Cotang      Tang 

1           61°                        60°                        59°            ll            68° 

TABLE  XXI.— TANGENTS  AND  COTANGENTS. 


355 


32°                       33°                       34°                       35° 

Tang     Cotang   i   Tang 

Cotang 

Tang     Cotang      Tang 

Cotang 

0 

.62487 

1.600->!        .«f:»  11 

1.53986 

.67451 

1.48256 

.70021 

1.42815 

60 

1 

.62527 

1.51  »'.»;  JO        .04982 

1.53888 

.67493 

1.48163 

.70004 

1.42720 

59 

2 

.62568 

1.59820  ll    .65024 

1.53791 

.67536 

1.48070 

.  70107 

1.42638 

58 

3 

.62608 

1.59723 

.05005 

1.53093 

.67578 

1.47977  ! 

.70151 

1.42550 

57 

4 

.62649 

1.59020 

.65106 

1.53595 

.67620 

1.47885  i 

.70194 

1.42462 

56 

5 

.62689 

1.59517 

.65148 

1.53497 

.67663 

1.47792 

.70238 

1.42374 

55 

6 

.62730 

1.59414 

.65189 

1.53400 

.67705 

1.47699 

.70281 

1.42286 

54 

7 

.62770 

1.59311 

.65231 

1.53302 

.67748 

1.47607 

,70325 

1.42198 

53 

8 

.62811 

1.59208 

.65272 

1.53205 

.67790 

1.47514 

.70308 

1.42110 

52 

9 

.62852 

1.59105 

.65314 

1.53107 

.67832 

1.47422 

.70412 

1.42022 

51 

10 

.62892 

1.59002 

'.65355 

1.53010 

.67875 

1.47330 

.70455 

1.41934 

50 

11 

.629S3 

1.58900 

.65397 

1.52913 

.67917 

1.47238 

.70499 

1.41847 

49 

12 

.62973 

1.5871)7 

.65438 

1.52816 

.67900 

1.47146 

.70543 

1.41759 

48 

13 

.63014 

1.58606 

.65480 

1.52719 

.08002 

1.47053 

.705v°3 

1.41672 

47 

14 

.63055 

1.58593 

.65521 

1.52022 

.68045 

1.46962 

!rOG29 

1.41584 

46 

15 

.63095 

1.58490 

.65563 

1.52525 

.68088 

1.46870 

.70673 

1.41497 

45 

6 

.63136 

1.58388 

.65604 

1.52429 

.68130 

1.46778 

.70717 

1.41409 

44 

.63177 

1.58286 

.65646 

1.52332  ! 

.68173 

1.46686 

.70760 

1.41322 

43 

8 

.63217 

1.58184 

.65688 

1.52235  i 

.68215 

1.46595 

.70804 

1.41235 

42 

19 

.63258 

1.58083 

.65729 

1.52139  ! 

.68258 

1.46503 

.70848 

1.41148 

41 

20 

.63299 

1.57981 

.65771 

1.52043  ; 

.68301 

1  .4647-1 

.70891 

1.41061 

40 

21 

.6.3340 

1.57879 

.65813 

1.51946 

.68343 

1.46320 

.70935 

1.40974 

39 

22 

.63380 

1.57773 

.65854 

x.  51850 

.08386 

1.46229 

.70979 

1.40887 

38 

23 

.63421 

1.57676 

.65896 

1.51754 

.68429 

1.46137 

.71023 

1.40800 

37 

24 

.63462 

1.57575 

.65938 

1.51058 

.68471 

1.46046 

.71066 

1.40711 

36 

25 

.63503 

1.57474 

.65980 

1.51562 

.68514 

1.45955 

.71110 

1.40027 

35 

26 

.63544 

1.57372 

.66021 

1.51466  i 

.68557 

1.45864 

.71154 

1.40.540 

•;.4 

27 

.63584 

1.678*1 

.66063 

1.51370  | 

.68600 

1.45773 

.71198 

1.40454 

33 

28 

.63625 

1.57170 

.66105 

1.51275 

.68642 

1.45682 

.71242 

1.40367 

32 

2U 

.63666 

1.57069 

.66147 

1.51179 

.68685 

1.45592 

.71285 

1.40281 

31 

30 

.63707 

1.56969 

.66189 

1.51084  ! 

.68728 

1.45501 

.71329 

1.40195 

30 

31 

.63748 

1.56868 

.66230 

1.50988 

.68771 

1.45410 

.71373 

1.40109 

29 

32 

.63789 

1.56767 

.66272 

1.50893 

.68814 

1.45320 

.71417 

1.40022 

28 

33 

.63830 

1.56007 

.66314 

1.50797 

.68857 

1.45229 

.71461 

1.89936 

27 

34 

.63871 

1.56566 

.66356 

1.50702 

.68900 

1.45139 

.71505 

1.39850 

26 

35 

.63912 

1.56466 

.66398 

1.50607 

.68942 

1.45049 

.71549 

1.39764 

25 

36 

.63953 

1.56366 

.66140 

1.50512 

.68985 

1.44958 

.71593 

1.39679 

24 

37 

.63994 

1.56265 

.66482 

1.50417 

.69028 

1.44868 

.71637 

1.39593 

23 

38 

.64035 

1.56165 

.00524 

1.50322 

.09071 

1.44778 

.71681 

1.39507 

22 

39 

.64076 

1.5GOG5 

.66506 

1.50228 

.69114 

1.44688 

.71725 

1.39421 

21 

40 

.64117 

1.55966 

.60608 

1.50133 

.69157 

1.44598 

.71769 

1.39336 

20 

41 

.64158 

1.55866 

.66650 

1.50038 

.69200 

1.44508 

.71813 

1.39250 

19 

42 

.64199 

1.55766 

.66692 

1.49944 

.69243 

1.44418 

.71857 

1.39165 

18 

43 

.64240 

1.55666 

.66734 

1.49849 

.69286 

1.44329 

.71901 

1.39079 

17 

44 

.64281 

1.55567 

.06776 

1.49755 

.69329 

1.44239 

.71946 

1.38994 

16 

45 

.64322 

1.55467 

.66818 

1.49661 

.6937'2 

1.44149 

.71990 

1.38909 

15 

46 

.64363 

1.55368 

.86860 

1.49566 

.69416 

1.44060 

.72034 

1.38824 

14 

47 

.64404 

1.55269 

.66902 

1.49472 

.69459 

1.43970       .72078 

1.38738 

13 

48 

.64446 

1.55170 

.66944 

1.49378 

.09502 

1.43881 

.72122 

1.38653 

12 

49 

.64487 

1.55071 

.66986 

1.49284 

.69545 

1.43792 

.72167 

1.38568 

11 

50 

.64528 

1.54972 

.67028 

1.49190 

.69588 

1.43703 

.72211 

1.38484 

10 

51 

.64569 

1.54873 

.67071 

1.49097 

.69631 

1.43614 

.72255 

1.88399 

9 

52 

.64610 

1.54774 

.67113 

1.49003 

.69675 

1.43525 

.7'2299 

1.38314 

8 

53 

.64652 

1.54675 

!    .67155 

1.48909 

.69718 

1.43436 

.72344 

1.38229 

54 

.64693 

1.54576 

i   .67197 

1.48816 

.69761 

1.43347 

.72388 

1.38145 

6 

55 

.64734 

1.54478 

.67239 

1.48722 

.69804 

1.43258 

.72432 

1.38060 

P 

56 

.64775 

1.54379 

!   .67282 

1.48629 

.69847 

1.43169 

.72477 

1  37976 

i 

5? 

.64817 

1.54281 

i   .67324 

1.4R536 

.69891 

1.43080 

.72521 

1.37891 

t 

58 

.64858 

1.54183 

i   .67366 

1.48442 

.69934 

1.42992 

.72565 

1  37807 

5£ 

.64899 

1.54085 

1    .67409 

1.48849 

.69977 

1.42903 

.72610 

1.37722 

• 

6C 

.64941 

1.53986 

.67451 

1.48256       .70021 

1.42815 

.72654 

1.37638 

( 

/ 

Cotang     Tang 

Cotang 

Tang     i  Cotang  j    Tang 

Cotang 

Tang 

i 

57° 

56°                      55°           II           64° 

TABLE  XXI.— TANGENTS   AND   COTANGENTS. 


36°                       37°            i           88°           ||           39° 

Tang  |  Cotang  |j  Tang  j  Cotang 

Tang  i  Cotang  il   Tang     Cotang 

0 

.7^654 

1.37638  !    .75355" 

l.r->704 

.78129 

I  .27994 

.80978 

1.23490 

60 

1 

.72099 

1.37554 

.75401 

l.JfcttWl 

.78175 

1.27917 

.81027 

1.23416 

59 

2 

.72743 

1.37470 

.75447 

1.32544 

.78228 

1.27841 

.81075 

1.23343  ;58 

3  1   .72788 

1.37386 

.75493 

1.32464 

.78209 

1.27704 

.81123 

1.23270 

57 

4 

72832 

1.37302 

.75538 

1.32384 

.78316 

1.27688 

.81171 

1.23196 

56 

5 

.72877 

1.37218 

.75584 

1.32304 

.78303 

1.27611 

.81220 

1.23123 

55 

6 

.72921 

1.37134 

.75029 

1.32224 

.78410 

1.27535 

.81268 

1.23050 

54 

7 

.72906 

1.37050 

.75075 

1.32144 

.78457 

1.27458 

.81316 

1.22977 

53 

8 

.73010 

1.30967 

.75721 

1.32064 

.7'8504 

1.27383 

.81364 

1.22904 

52 

9 

.73055 

1.30883 

.75767 

1.31984 

.78551 

1.27306 

.81413 

1.22831 

51 

10 

.73100 

1.36800 

.75812 

1.31904 

.78598 

1.27230 

.81461 

1.22758 

50 

11 

.73144 

1.36716 

.75858 

1.31825 

.78645 

1.27153 

.81510 

1.22685 

49 

12 

.73189 

1.36633 

.75904 

1.31745 

.78698 

1.27077 

.81558 

1.22012 

48 

13 

.73234 

1.36549 

.75950 

1.31666 

.78739 

1.27001 

.81606 

1.22539 

47 

14 

.73278 

1.  30466 

.75996 

1.31586 

.78786 

1.20925 

.81655 

1.  22407 

46 

15 

.73323 

1.36383 

.76042 

1.31507 

.78834 

1.26849 

.81703 

1.22394 

45 

16    .73368 

1.30300 

.76088 

1.31427 

.78881 

1.26774 

.81752 

1.22321 

44 

17    .73413 

1.30217  i 

.76134 

1.31348 

.78928 

1.26698 

.81800 

1/22249 

43 

18     .73457 

1.30134  i 

.76180 

1.31209 

.78975 

1.26622 

.81849 

1.22178 

•JO 

19'    .73503 

1.30051 

.76226 

1.31190  i 

.79022 

1.26546 

.81898 

1.22104 

41 

20 

.73547 

1.35968 

.76272 

1.31110 

.79070 

1.26471 

.81946 

1.22031 

40 

21 

.73592 

1.35885 

.76318 

1.31031 

.79117 

1.26395 

.81995 

1.21959 

39 

22 

.73637 

1.33803  ! 

.76304 

1.30952 

.79104 

1.26319 

.82044 

1.21886 

38 

23 

.73681 

1.35719  ! 

.76410 

1.30873 

.79212 

1.26244 

.82092 

1.21814 

37 

24 

,73726 

1.35037 

.76456 

1.30795 

.79259 

1.26169 

.82141 

1.21742 

36 

25 

.73771 

1.35554 

.76502 

1.30716 

.79306 

1.26093 

.82190 

1.21670 

35 

26 

.73816 

1.35472 

.76548 

1.30637 

.79354 

1.26018 

.82238 

1.21598 

34 

27 

.73861 

1.35389 

.76594 

1.30558 

.79401 

1.25943 

.82287 

1.21526 

33 

28 

.73906 

1.35307  i 

.76640 

1.30480 

.79449 

1.25867 

.82336 

1.21454 

32 

29 

.73951 

1.352^4 

.76686 

1.30401 

.79496 

1.25792 

.82385 

1.21382 

31 

30 

.73996 

1.35142  i 

.76733 

i.30323 

.79544 

1.25717 

.82434 

1.21310 

30 

31 

.74041 

1.35060  ' 

.76779 

1.30244 

.79591 

1.25642 

.824&3 

1.21238 

29 

32 

.74086 

1.34978 

.76825 

1.30166 

.79639 

1.25567 

.82531 

1.21166 

28 

33 

.74131 

1.34806 

.70871 

1.30087 

.79686 

1.25492 

.82580 

1.21094    27 

34 

.74176 

1.34814 

.70918 

1.30009 

.79734 

1.25417 

.82629 

1.21023 

26 

35 

.74221 

1.34732 

.70964 

1.29931 

.79781 

1.25343 

.82678 

1.20951 

25 

36 

.74207 

1.34650 

.77010 

1.29853  1 

.79829 

1.25268 

.82727 

1.20879 

24 

37 

.74312 

1.34568 

.77057 

1.29775 

.79877 

1.25193 

.82776 

1.20808 

23 

38 

.74357 

1.34487 

.77103 

1.29696 

.79924 

1.25118 

.82825 

1.20736 

22 

39 

.71402 

1.34405 

.77149 

1.29618 

.79972 

1.25044 

.82874 

1.20605 

21 

40 

.7414!? 

1.34323 

.77196 

1.29541 

.80020 

1.24969 

.82923 

1.20593 

20 

41 

.74492 

1.34242 

.77243 

1.29463 

.80067 

1.24895 

.82972 

1.20522 

19 

42 

.74538 

1.34160 

.77289 

1.29385 

.80115 

3.24820 

.83022 

1.20451 

18 

43 

.74583 

1.34079 

.77335 

1.29307 

.80163 

1.24746 

.83071 

1.20379 

17 

44 

.74628 

1.33998 

.77382 

1.29229 

.80211 

1.21672 

.83120 

1.20308 

16 

45 

.74674 

1.33916 

.77428 

1.39152 

.80258 

1.24597 

.83169 

1.20237 

15 

46 

.74719 

1.33835 

.77475 

1.29074 

.80306 

1.24523 

.83218 

1.2016G 

14 

47 

.74764 

1.33754 

.77521 

1.28997 

.80354 

1.24449 

.83268 

1.20095 

13 

48 

.74810 

1.33673 

.77568 

1.28919 

.80402 

1.24375 

.83317 

1.20024 

12 

49 

.74855 

1.33592 

.77615 

1.28842 

.80450 

1.24301 

.83366 

1.19953    11 

50 

.74900 

1.33511 

.77661 

1.28764 

.80498 

1.24227 

.83415 

1.19882 

10 

51 

.74946 

1.33430 

.77708 

1.28687 

.80546 

1.24153 

.83465 

1.19811 

9 

52 

.74991 

1.33349 

.77754 

1.28610 

.80594 

1.24079 

.83514 

1.19740 

8 

53 

.75037 

1.33268 

.77801 

1.28533 

.80642 

1.24005 

.83564 

1.19609 

7 

54 

.75082 

1.33187 

.77848 

1.28456 

.80690 

1.23931 

.83613 

1  .  19599 

6 

55 

.75128 

1.33107 

,77895 

1.28379 

.80738 

1.23858 

.83662 

1.19528 

5 

56 

.75173 

1.33026 

.77941 

1.28302 

.80786 

1.23784 

!  83712 

1.19457 

4 

57 

.75219 

1.32946 

.77938 

1.28225 

[80884 

1.23710 

.83761 

1.19387 

3 

58 

.75264 

1.32865 

.78035 

1.28148 

.80882 

1.23637 

.88811 

1  .  19316 

2 

59 

.75310 

1.32785 

.78082 

1.28071 

.80930 

1.23563 

.83860     1.19246 

1 

60 

.75355 

1.32704 

.78129 

1.27994 

.80978 

1.23490 

.831/10  ;  1.19175 

0 

I 

Cotang 

Tang 

Cotang 

Tang 

Cotang     Tang    ','  Cotang     Tang 

/ 

53° 

52° 

51°                      60° 

TABLE  XXI.-TANGENTS   AND  COTANGENTS. 


4 

0° 

4 

1° 

4 

2° 

4 

3° 

Taiig 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.83910 

1.19175 

.86929 

1.15037 

.90040 

1.11061 

.93252 

1.07237 

60 

1 

.83960 

1.19105 

.86980 

1.14969 

.90093 

1.10996 

.93306 

1.07174 

59 

9 

.84009 

1.19035 

.87031 

1.14902 

.90146 

1.10931 

.93360 

1.07112 

58 

a 

.84059 

1.18964 

.87082 

1.14834 

.90199 

1.10867 

.93415 

1.07049 

57 

4 

.84108 

1  .  18894 

.87133 

1.14767 

.90251 

1.10802 

.93469 

1.06987 

56 

6 

.84158 

1.18824 

.87184 

1.14699 

.90304 

1.10737 

.93524 

1.06925 

55 

C 

.84208 

1.18754 

.87236 

1.14632 

.90357 

1.10672 

.93578 

1.06862 

54 

7 

.84258 

1.18684 

.87287 

1.14565 

.90410 

1.10607 

.93633 

1.06800 

53 

8 

.84307 

1.18614 

.87338 

1.14498 

.90463 

1.10543 

.93688 

1.06738 

52 

i> 

.84357 

1  .  18544 

.87389 

1.14430 

.90516 

1.10478 

.93742 

1.06676 

51 

10 

.84407 

1.18474 

.87441 

1.14363 

.90569 

1.10414 

.93797 

1.06613 

50 

11 

.84457 

1.18404 

.87492 

1.14296 

.90621 

1.10349 

.93852 

1.06551 

41) 

12 

.84507 

1.18334 

.87543 

1.14229 

.90674 

1.10285 

.93906 

1.06489 

48 

13 

.84556 

1  .  18264 

I   .87595 

1.14162 

.90727 

1.10220 

.93961 

1.06427 

4. 

14 

.84606 

1.18194 

.87646 

1.14095 

.90781 

1.10156 

.94016 

1.06365 

16 

15 

.84656 

1  .  18125 

!   .87698 

1.14028 

.90834 

1.10091 

.94071 

1.06303 

15 

16 

.84706 

1.18055 

.87749 

1.13961 

.90887 

1.10027 

.94125 

1.06241 

44 

17 

.84756 

1.17986 

.87801 

1.13894  ' 

.90940 

1.09963 

.94180 

1.06179 

13 

18 

.84806 

1.17916 

.87852 

1.13828  : 

.90993 

1.09899 

I   .94235 

1.06117 

I'.! 

19 

.84856 

1.17846 

!   .87904 

1.13761 

.91046 

1.09834 

.94290 

1.06056 

11 

20 

.84906 

1.17777 

;    .87955 

1.13694 

.91099 

1.09770 

.94345 

1.05994 

10 

21 

.84956 

1.17708 

.88007 

1.13627 

.91153 

1.09706 

.94400 

1.05932 

39 

22 

.85006 

1.17638 

.88059 

1.13561 

.91206 

1.09642 

.94455 

1.05870 

88 

23 

.85057 

1.17569 

i   .88110 

1.13494 

.91259 

1.09578 

.94510 

1.05809 

87 

2-1 

.85107 

1  .  17500 

.88162 

1.13428 

.91313 

1.09514 

.94565 

1.05747 

80 

25 

.85157 

1.17430 

.88214 

1.13361 

.91366 

1.09450 

.94620 

1.05685 

35 

26 

.85207 

1.17361 

.88265 

1.13295 

.91419 

1.09386 

.94676 

1.05624 

34 

27 

.85257 

1.17292 

.88317 

1.13228 

.91473 

1.09322 

.94731 

1.05562 

33 

28 

.85308 

1.17223 

.88369 

1.13162 

.91526 

1.09258 

.94786 

1.05501 

32 

29 

.85358 

1.17154 

.88421 

1.13096 

.91580 

1.09195 

.94841 

1.05439 

81 

30 

.85408 

1.17085 

.88473 

1.13029 

.91633 

1.09131 

.94896 

1.05378 

30 

31 

.85458 

1.17016 

.88524 

1.12963 

.91687 

1.09067 

.94952 

1.05317 

29 

32, 

.85509 

1.16947 

.88576 

1.12897 

.91740 

1.09003 

.95007 

1.05255 

28 

33 

.85559 

1.16878 

.88628 

1.12831 

.91794 

1.08940 

.95062 

1.05194 

27 

34 

.85609 

1.16809 

'.88680 

1.12765 

.91847 

1.08876 

.95118 

1.05133 

26 

35 

.85660 

1.16741 

.88732 

1.12699 

.91901 

1.08813 

.95173 

1.05072 

25 

36 

.85710 

1.16672 

.88784 

1.12633  I 

91955 

1.08749 

I   .95229 

1.05010 

24 

37 

.85761 

1.16603 

.88836 

1.12567 

.92008 

1.08686 

i   .95284 

1.04949 

23 

38 

.85811 

1.16535 

.88888 

1.12501  : 

.92062 

1.08622 

.95340 

1.04888 

23 

39 

.85862 

1.16466 

.88940 

1.12435  l 

.92116 

1.08559 

.95395 

1.04827 

-,'1 

40 

.85913 

1.16398 

.88992 

1.12369 

.92170 

1.08496 

.95451 

1.04766 

20 

41 

.85963 

1.16329 

.89045 

1.12303 

.92224 

1.08432 

.95506 

1.04705 

1!) 

42 

.86014 

1.16261 

.89097 

1.12238 

.92277 

1.08369 

.95562 

1.04644 

18 

43 

.86064 

1.16192 

.89149 

1.12172 

.92331 

1.08306 

.95618 

1.04583 

17 

44 

.86115 

.16124 

.89201 

1.12106 

.92385 

1.08243 

.95673 

1.04522 

16 

45 

.86166 

.16056 

.89253 

1.12041 

.92439 

1.08179 

.95729 

1.04461 

15 

46 

.86216 

.15987 

.89306 

1.11975 

.92493 

1.08116 

.95785 

1.04401 

14 

47 

.86267 

.15919 

.89358 

1.11909 

.92547 

1.08053 

.95841 

1.04340 

13 

48 

.86318 

.15851 

.89410 

1.11844 

.92601 

1-07990 

.95897 

1.04279 

13 

49 

.86368 

.15783 

.89463 

1  11778 

.92655 

1.07927 

.95952 

1.04218 

11 

50 

.86419 

.15715 

.89515 

1.11713 

.92709 

1.07864  : 

.96008 

1.04158 

10 

51 

.86470 

.15647 

.89567 

1.11648 

.92763 

1.07801  ! 

.96064 

1.04097 

9 

52 

.86521 

.  15579 

.89620 

1.11582 

.92817 

1.07738  i 

.96120 

1.04036 

8 

53 

.86572 

.15511 

.89672 

1.11517 

.92872 

1.07676  ! 

.96176 

1.03976 

7 

54 

.86623 

.15443 

.89725 

1.11452 

.92926 

1.07613  i 

.96232 

1.03915 

6 

55 

.86674 

.15375 

.89777 

1.11387 

.92980 

1.07550 

.96288 

1.03865 

5 

56 

.86725 

.15308 

.89830 

1.11321 

.93034 

1.07487  i 

.96344 

1.03794 

4 

57 

.86776 

.15240 

.89883 

1.11256 

.93088 

1.07425 

.96400 

1.03734 

3 

58 

.86827 

.15172 

.89935 

1.11191 

.93143 

1.07362 

.96457 

1.0:3674 

L' 

59 

.86878 

1.15104 

.89988 

1.11126 

.93197 

1.07299 

.96513 

1.03613 

1 

00 

.86929 

1.15037 

.90040 

1.11061 

.93252 

1.07237 

.96509 

1.03553 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

i 

4 

9° 

4 

8° 

i           4 

7° 

4 

6° 

TABLE  XXI.— TANGENTS   AND   COTANGENTS. 


4 

4° 

4 

4°               ,  II 

4 

4° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.96569 

.03553 

60 

20 

.97700 

1.02355  -40  — 

0  i     .98843 

1.01170 

20 

1 

.96625 

.03493 

59 

21 

.97756 

1.02295     39     - 

1        .5)8901 

1.01112 

19 

2 

.96681 

.03433 

58 

22 

.97813 

1.02236  '  38     - 

2  :      .'.)*«.  )5S 

1.01053 

18 

3 

.96738 

.03372 

57 

23 

.97870 

1.02116     37     ' 

3     .won; 

1.00994 

17 

4 

.96794 

.03312 

56 

24 

.97927 

1.02117  ,36     H 

I        .1KK173 

1.00935 

16 

6 

.96850 

.03252 

55 

25 

.97984 

1.02057  jay1  4 

5       .99131 

1.00876 

15 

i    6 

.96907 

.03192 

54 

26 

.98041 

1.01998     34     -J 

.-;     .ii;ii8;) 

1.',  OS  IS 

14 

7 

.96963 

.03132 

53 

27 

.98098 

1.011)31)    33     4 

7       .91)24? 

1.00759 

13 

8 

.97020 

.03072 

53 

28 

.98155 

1.01879    :w 

8       .99304 

1.00701 

12 

0 

.97076 

.03012 

51 

29 

.98213 

1.01820    31:14 

9       .99362 

1.00642 

11 

10 

.97133 

.02952 

50 

30 

.98270 

1.01761     30    £ 

0       .99-420 

1.00583 

10 

11 

.97189 

.02892 

49 

31 

.98327 

.01702    29!  'f 

1       .99478 

1.00525 

9 

.97246 

.02832 

48 

32 

.98384 

:  .01642     2H     f 

1.00467 

8 

13 

.97302 

.02772 

47 

83 

.98441 

.01583     27     F 

3       .99594 

1.00108 

7 

14 

.97359 

.02713 

46 

34 

.98499 

.01524     26     T 

4       .9965  J 

1    ~>0')50 

6 

15 

.97416 

.02653 

45 

85 

.98556 

.01465     25     T 

5        .W710 

1.  W291 

5 

16 

.97472 

.02593 

44 

36 

.98613 

.01406     24     T 

8       .99768 

1  .  10233 

4 

17 

.97529 

.02533 

43 

37 

.98671 

:  .01347     23     f 

7       .99826 

1.  )0  175 

3 

18 

.97586 

.02474 

42 

38 

.98728 

.012S8     22     f 

8"      .99884 

1.10116 

19 

.97643 

.02414 

41 

39 

.98786 

.01229     21    ifi 

!)       .  !Ht:M. 

1     'Oii.i,*. 

T 

20 

.97700 

.02355 

40 

40 

.98843 

.01170     20:   ( 

0  |  l.OOOUO 

1  .  HKH'M) 

0 

Cotang 

Tang 

Cotang 

Tang     j 

Cotang 

Tang 

4 

5° 

I 

4 

5°           1       1 

4 

5° 

J 

TABLE   XXII.-VKRSIXES   AND   KXS 


' 

0°          1° 

2° 

3° 

' 

Vers.  !  Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec, 

0 

.00000   .00000 

.00015 

.00015 

.00061 

.00061 

.00137 

.00137 

0 

1 

.00000  ;  .00000 

.00016 

.00016 

.00062 

.00062 

.00139 

00139 

1 

2  1  .00000 

.00000 

.00016 

.00016 

.00063 

.00063 

.00140 

.00140 

2 

3   .00000 

.00000 

.00017 

.00017 

.00064 

.00064 

.00142 

.00142 

3 

4 

.00000 

.00000 

.00017 

.00017 

.00065 

.00005   .00143 

.00143 

4 

5 

.00000 

.0)000  (1  .00018 

.00018 

.00066 

.00066 

.00145 

.00145 

5 

6 

.00000 

.00000  1  .00018 

.00018 

.00067 

.00067 

.00146 

.00147 

6 

7 

.00000 

.00000 

.00019 

.00019 

.00068 

.00068 

.00148 

.00148 

7 

8 

.00000 

.00000 

.00020 

.00020 

.00069 

.00069 

.00150 

.00150 

8 

9 

.00000 

.00000 

.00020 

.00020 

.00070 

.00070 

.00151 

.00151 

9 

to 

.00000 

.00000 

.00021 

.00021 

.00071 

.00072 

.00153 

.00153 

10 

11 

.00001 

.00001 

.00021 

.00021 

.00073 

.00073 

.00154 

.00155 

11 

.00001 

.00001 

.00022 

.00022 

.00074 

.00074 

.00156 

.00156 

12 

13 

.00001 

.00001 

.00023 

.00023 

.00075 

.00075 

.00158 

.00158 

13 

14 

.00001  !  .00001 

.00023 

.00023 

.00076 

.00076 

.00159 

.00159 

14 

15 

.00001   .00001 

.00024 

.00024 

.00077 

.00077 

.00161 

.00161 

15 

16 

.00001 

.00001 

.00024 

.00024 

.00078 

.00078 

.00162 

.00163 

16 

17 

.00001 

.00001 

.00025 

.00025 

.00079 

.00079 

.00164 

.00164 

17 

18 

.00001 

.00001 

.00026 

.00026 

.00081 

.00081 

.00166 

.00166 

18 

19 

.00002 

.00002 

.00026 

.00026 

.00082 

.00082 

.00168 

.00168 

19 

20   .00002 

.00002 

.00027 

.00027 

.00083 

.00083 

.00169 

.00169 

20 

21 

.00002 

.00002 

.00028 

.00028 

.00084 

.00084 

.00171 

.00171 

21 

22 

.00002 

.00002 

.00028 

.00028 

.00085 

.00085 

.00173 

.00173 

22 

£3 

.00002 

.00002 

.00029 

.00029 

.00087 

.00087 

.00174 

.00175 

23 

24 

.00002 

.00002 

.00030 

.00030 

.00088 

.00088 

.00176 

.00176 

24 

85 

.00003 

.00003 

.00031 

.00031 

.00089 

.00089 

.00178 

.00178 

25 

26   .00003 

.00003 

.00031 

.00031 

.00090 

.00090 

.00179 

.00180 

27   .00003 

.00003 

.00032 

.00032 

.00091 

.00091 

.00181 

.00182 

H~ 

28   .00003 

.00003 

.00033 

.00033 

.00093 

.00093 

.00183 

.00183 

28 

29   .00001 

.00004 

.00031 

.00034 

.00091 

.00094 

.00185 

.00185 

29 

30   .00004 

.00004 

.00034 

.00034 

.00095 

.00095 

.00187 

.00187 

30 

31 

.00001 

.00004 

.00035 

.00035 

.00096 

.00097 

.00188 

.00189 

31 

32 

.00004 

.00004 

.00036 

.00036 

.00098 

.00098 

.00190 

.00190 

32 

83 

.00005 

.00005 

.00037 

.00037 

.00099 

.00099 

.00192 

.00192 

33 

34 

.00005 

.00005 

.00037 

.00037 

.00100 

.00100 

.00191 

.00194 

34 

35 

.00005 

.00005 

.00038 

.00038 

.00102 

.00102 

.00196 

.00196 

35 

36 

.00005 

.00005 

.00039 

.00039 

.00103 

.00103 

.00197 

.00198 

36 

37 

.00006 

.00006 

.00040 

.00040 

.00104 

.00101 

.00199 

.00200 

37 

38 

.00006 

.00006 

.00041 

.00041 

.00106 

.00106 

.00201 

.00201 

38 

39 

.00006 

.00006 

.00041 

.00041 

.00107 

.00107 

.00203 

.00203 

39 

40 

.00007 

.00007 

.00042 

.00042 

.00108 

.00108 

.00205 

.00205 

40 

41   .00007 

.00007 

.00043 

.00043 

.00110 

.00110 

.00207 

.00207 

41 

42   .00007 

.00007  \ 

.00014 

.00044 

.00111 

.00111 

.00208 

.00200 

42 

43 

.00008 

.00008 

.00045 

.00045 

.00112 

.00113 

.00210 

.00211 

43 

44 

.00008 

.00008 

.00016 

.00046 

.00114 

.00114 

.00212 

.00213 

44 

46 

.00009 

.00009 

.00047 

.00047 

.00115 

.00115 

.00214 

.00215 

45 

46 

.00009 

.00009 

.00018 

.00048 

.00117 

.00117 

.00216 

.00216 

46 

47 

.00009 

.00009 

.00048 

.00048 

.00118 

.00118 

.00218 

.00218 

47 

48 

.00010 

.00010 

.00019 

.00049 

.00119 

.00120 

.00220 

.00220 

48 

49 

.00010 

.00010 

.00050 

.00050 

.00121 

.00121 

.00222 

.00222 

49 

50 

.00011 

.00011 

.00051 

.00051 

.00122 

.00122 

.00224 

.00224 

50 

51 

.00011 

.00011 

.00052 

.00052 

.00124 

.00124 

.00226 

.00226 

51 

52 

.00011 

.00011 

.00053 

.00053 

.00125 

.00125 

.00228 

.00228 

5.0 

53 

.00012 

.00012 

.000.54 

.00054 

.00127 

.00127 

.00230 

.00230 

53 

54 

.00012 

.00012 

.00055 

.00055 

.00128 

.00128 

.00232 

.00232 

54 

55 

.00013 

.00013 

.00056 

.00056  |  .00130 

.00130 

.00234 

.00231 

55 

56 

.00013 

.00013 

,00057 

.00057  I  .00131 

.00131 

.00236 

.00236 

57 

.00014 

.00014 

.00058 

.00058  I!  .00133 

.00133 

.00238 

.00338 

57 

58 

.00014 

.00014 

.00059 

.00059   .00134 

.00134 

.00240 

.00240 

58 

59  I  .00015 

.00015 

.00060 

.00060 

.00136 

.00133 

.00212 

.00242 

59 

60  1  .00015 

.00015 

.00061 

.00061  i 

.00137 

.00137  1 

.00244 

.00244 

60 

TABLE   XXII.     VEK.SINES    AND    KXSKCANTS, 


12° 

1 
13° 

14° 

15° 

Vers.     Exsec.       Vers.     Exsec.    .   Vers.     Exsec.      Yers.     Ex?ec. 


.(1-2185 
.02191 
.02197 

.  ea  j 

.02210 
.02216 


.  •.-.  ; 
.02340 
.02246 

.02253 


.02271 
.02277 


.02314 

j  - !   . 


.02364 
.02370 

.02377 


.  $429 
.02434 

.02440 
.02447 
.•  J453 


.02472 

.03479 


.1  3493 


.02504 
.02511 
.08517 


.06530 
.03695 
.  3548 
.02550 


.02834 

.02340 
.02247 
.02253  I 
!  02258 
.02266 
.02272 
.02279 
.0-2285  ! 
.02291 
.03398  , 

.02304  | 
.02:311 
.02317  ! 
.02323- 
.03330  \ 

.02343 
.02349 

.  -..  56 


.02:369 

.1  ,-.  " 


.  ••-::  3 
.02415 
.02421 


.02441 


.02454 
.02461 


.02474 


.03509 


.02515 
.•  2521 


.  3548 
.02555 


.03560 

.02570 
.02576 


.02616 
.02623 


.02642 

.02649 
.02655 
.1  .  H 


.02!  • 
.02716 
.03722 
.03729 

.  .; . 

.02743 


.'  B56 


.02770 
.02777 
!l  3788 
[02790 


02569  i     .03907 

02576       .02914 

.02921 


.03589 
.03596 


.096  1 


.02928 
.02935 
.02943 


.02630 

.02637 
.02644 
.02651  ' 


.•  :  165 

.  8673 

.02679 
.C26S6 


.<  J834 


.  2870 


.•  . .  : 


.02987 
.02994 


.02970 


.03046 

•  •': 


8970 

.02977 


.  :,.-  0 

09  19 


.02700 
.02707  '] 

.02728 
.  •:'  « 
.(  3749 
.02749 
.02756 
.•  •->-, 
.03770 

.02777 
.02784 

.-  .;  -1 
.  ,-  6 


.03027 
.1  90M 
.03041 

.03048 

:; 
.    070 


.090  1 


.03106 
.03113 

.03120 
.0312! 


.03142 
.  H4S 


.03163 
.03171 
.03178 
.03185 
.03193 

.  .;  • 


/    814 


.1  3244 

.    ./I 
08258 


.(  879 


.0    .  ) 


.03076    ! 
.03084    | 


.03108 

.03114 
.03121 
.03129 
.03137 

.03144 
.03152 
.03159 
.03167 
.03175 

!08190 

.1  .'  18 


.a  aa 


.03244  i 
.08251  ! 


.   :••: 
.03275 


.03321 


.  8337 
.03345 
.033.33 


.03061  |,  .03-107 


.08416 
.08424 

.03432 
.03439 
.03447 

.03340  .03455 
!  .03463 
i  .03471 
.03479 
.03487 
.03495 
03503 
.08512 
.03920 


.03370 


.03400 


.03407 
.03415 
.03433 
.034:30 
.03438 
.03445 
.03453 


.03476 
.03483 

.03491 
.(  M98 
.03506 
.  3514 

.03529 
.03537 


.03560 
.03567 

.  /..  * 
.03598 
.08006 
.03614 
.03621 

.03637 
.03645 


.03676 


.03707 
.03715 
.03723 
.03731 
.03739 

.03762 
.03770 
[03778 
[03788 
.€0791 


.08810 
[03818 


.06842 


.08868 
!03B74 


ANTS. 


363 


16° 


17° 


18° 


Vers.     Exsec.       Vers.  ;  Exsec.  ;   Vers.     Exsec. 


to 
n 

i-J 
18 
14 
15 
16 
17 
18 
•.' 
U 


34 

35 

•:•', 
•  : 
88 
89 
40 

41 

-I-.! 

43 
44 
45 
46 
47 
48 
49 
60 

M 


.088T4  i   .04030 


.03914  i 
.03953 


.03071 


.04011  I 
.04019  ! 
.    1088 
.04036  | 

.04044  i 

.•     :     '      i 


.04085 
.04  03 


.04110 
.04118 

.04126 
.04135 
.04143 
.04151 
.04159 
[04168 
.04176 


0*193 

.04501 


.04334 
.1  1868 


.04303 
.04810 
[04819 
.04337  j 
.0483  : 
.<  1844 

.043(51 


.04047 

.04056 

.04065 

.04073 

.04083 

.04091 

.04100    I 

.04108 

.04117 

.04136 
.04135 
.04144 

.04161 
.04170 
.04179 
.04188 

.04306 

.04314    i 

.04333 

.04333    ! 

.04341 

.04350 

.04359 

41968 

.04377 

.04395 

.04304 
.04313 
.04388 
[04331 
.04:340 
.04349 
04S58 
.04867 
.04376 
.04385 

.04394* 

.04403 

.04413 

.04433 

.04431 

.04440 

.04449 

.04458 

.04468 

.04477  j 

.04486  j 

.04496 

.04504 

.04514 

.04583 

.04589 

.04541 

.04551 


.04370 
.04378 
.04387 
.04395 
.04404 
.04413 
.04431 
.04420 
.04438 
.04446 
.04455 

.04464 
.04473 
.04481 


.04493 
.04507 
.04515 
.04534 


.04541 

.04550 
.04559 
44567 
.04576 
[04585 
.04593 
44602 
.04611 


.04569 
.04578 
.04588 
.04597 
.04606 
.<  1616 
.04635 
.04635 
.04644 
.04653 
.04663 

.04683-  j 


.04894  i 
.04903  ; 
.04913 
.04931 
!04980 


.04948 
.04957 


.05146 
.05156 
.05166 
.05176 
.05186 
.05196 
[05806 
.05816 


19' 


Vers.     Exsec. 


.05448 

.05458 
.05467 
.05477 
[05486 


.04637 
.04646 
.04655 
.04668 
[04673 
.04681 
[04600 


.04700 

.04710 

.04719 

.04739 

.04738 

.04748  , 

.04757 

.04767    I 

.04776 

.04786 

.04795 

.•  1805 

.04815 

.04834 
.04843 
.04853  ; 


.04976 
.04985  • 

.04994 
.05003  i 
.05013 
[05021 


.05346 
.06856 


.065  5 
.05515 
.05534 
.05534 
.05543 

.05553 
[06562 
.06572 


.66768 

.05773 
.06783 
[05794 
.05806 
[05815 


.'  :••  88 
.05048 
.05067 
.05067  j 
.05076  ! 

.05065 
.••  .  : 
.05103 
.05113 
.05133 
.05131 
.05140 
.05149 


.05886 
.0589! 

.05307 
.05317 
.0632! 
!05837 
.05347 

.05357 
!05367 
.05878 


.1  5691 

.05610 


.05630 

.<••",:  9 


.05847  8 

--- 

.05869  10 

.05879  !  11 

.05890  13 

.05901  13 

.05911  14 

.05933  16 

.05944  17 

.05955  !  18 

.05965  19 

.05976  20 


.05649     .05987 


.  0541  - 
.05418 
.05489 


.05678 
.05687 


.04879 


.04703 

.04716 

.04725 
.04734 
.04743 
.04753 
.04760 
[04769 
.04778 
44387 
.04796 
.04806 

.04814 


.04841 
.04860 
44858 


.(  1891 
[04901 
.04911 


.05168  : 

.05177  ! 
[05186 

.05195  i 

[osaos 

[05814 


.04940 
.0495J 

.04959 
.04979 


.05848 
45851 


.05870 


.05807 


.05!  18 
.05088 


.05047 

.05057 
.050  7 
.06077 
45087 
.05697 
.05107 
.05116 


.06844 
.05354 


.06873 


.04885  !   .05136 
.04894      .05146 


.05401 
.05410 
45480 


.05489 
!05448 


.05449 

.05460 
.05470 
.05480 
.05490 
[06601 
[05511 
.06581 


.05543 


.05573 
[06684 
.06604 
.05604 

[05615 


05636 
[05646 

.05657 

.05667 
[05678 


.05699 
.05709 
[05780 
.05730 
.05741 
.05751 
[05762 


.05716 
.05726 
.05736 

.05746 
.05755 
.05765 
.05775 
.06785 
[06794 
[06804 
[05814 


06020 

.06030 
.06041 
.06052 


.06074 


.05843 


.05863 
[05873 


.05912 


.06951 


.06971 


.05991 
.06001 
[06011 
[06681 


30 

31 

.06107 
.06118 
[06189 
.06140 
.06151 

.06173  38 
.06184  39 
.06195  40 


.06206 
.06817 


41 

48 

.06339  44 

.06860  4.-' 

[06861  -tf 

.06879  6 

46889  H 


GO 


.06306 
.06317 


.06373 


.06407 
.06418 


3G4 


TABLE  XXII.-VERSINES  AND  EXSECANTS. 


/ 

2( 

>° 

2] 

L° 

22 

0 

28 

0 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

~~0~ 

.06031 

.06418 

.06642 

.07115 

.07282 

.07853 

.07950 

.08636 

0 

1 

.06041 

.06429 

.(XH553 

.07126 

.07293 

.07866 

.07961 

.08649 

1 

2 

.06051 

.06440 

.06663 

.07138 

.07303 

.07879 

.07972 

.08663 

2 

3 

.06061 

.06452 

.06673 

.07150 

.07314 

.07892 

.07984 

.08676 

3 

4 

.06071 

.06463 

.06684 

.07162 

.07325 

.07904 

.07995 

.08690 

4 

5 

.06081 

.06474 

.06694 

.07174 

.07336 

.07917 

.08006 

.08703 

5 

6 

.06091 

.06486 

.06705 

.07186 

.07347 

,07930 

.08018 

.08717 

6 

7 

.06101 

.06497 

.06715 

.07199 

.07358 

070-43 

.08029 

.08730 

7 

8 

.06111 

.06508 

.06726 

.07211 

.07369 

.07955 

.08041 

.08744 

8 

9 

.06121 

.06520 

.06736 

.07223 

.07380 

.07968 

.08052 

.08757 

9 

10 

.06131 

.06531 

.06747 

.07235 

.07391 

.07981 

.08064 

.08771 

10 

11 

.06141 

.06542 

.06757 

.07247 

.07402 

.07994 

.08075 

.08784 

11 

12 

.06151 

.06554 

.06768 

.07259 

.07413 

.08006 

.08086 

.08798 

12 

13 

.06161 

.06565 

.06778 

.07271 

.07424 

.08019 

.08098 

.08811 

13 

14 

.06171 

.06577 

.06789 

.07283 

.07435 

.08032 

.08109 

.08825 

14 

15 

.06181 

.06588 

.06799 

.07295 

.07446 

.08045 

.08121 

.08839 

15 

16 

.06191 

.06600 

.06810 

.07307 

.07457 

.08058 

.08132 

.08852 

16 

IT 

.06201 

.06611 

.06820 

.07320  ! 

.07468 

.08071 

.08144 

.08866 

17 

18 

.06211 

.06622 

.06831 

.07332 

.07479 

.08084 

.08155 

.08880 

18 

19 

.06221 

.06634 

.06841 

.07344 

.07490 

.08097 

.08167 

.08893 

19 

20 

.06231 

.06645 

.06852 

.07356 

.07501 

.08109 

.08178 

.08907 

20 

21 

.06241 

.06657 

.06863 

.07368 

.07512 

.08122 

.08190 

.08921 

21 

22 

.06252 

.06668 

.06873 

.07380 

.07523 

.08135 

.08201 

,  08934 

22 

23 

.06262 

.06680 

.06884 

.07393 

.07534 

.08148 

.08213 

.08948 

23 

24 

.06272 

.06691 

.06894 

.07405 

.07545 

.08161 

.08225 

.08962 

24 

25 

.06282 

.06703 

.06905 

.07417 

.07556 

.08174 

.08236 

.08975 

25 

26 

.06292 

.06715 

.06916 

.07429 

.07568 

.08187 

.08248 

.08989 

26 

27 

.06302 

.06726 

.06926 

.07442 

.07579 

.08200 

.08259 

.09003 

27 

28 

.06312 

.06738 

.06937 

.07454 

.07590 

.08213 

.08271 

.09017 

28 

29 

.06323 

.06749 

.06948 

.07466 

.07601 

.08226 

.08282 

.09030 

29 

30 

.06333 

.06761 

.06958 

.07479 

.07612 

.08239 

.08294 

.09044 

30 

31 

.06343 

.06773 

.06969 

.07491 

.07623 

.08252 

.08306 

.09058 

31 

32 

.06353 

.06784 

.06980 

.07503 

.07634 

.08265 

.08317 

.09072 

32 

33 

.06363 

.06796 

.06990 

.07516 

.07645 

.08278 

.08329 

.09086 

33 

34 

.06374 

.06807 

.07001 

.07528 

.07657 

.08291 

.08340 

.09099 

34 

35 

.06384 

.06819 

.07012 

.07540 

.07668 

.08305 

.08352 

.09113 

35 

36 

.06394 

.06831 

.07022 

.07553 

.07679 

.08318 

.08364 

.09127 

36 

37 

.06404 

.06843 

.07033 

.07565 

.07690 

.08331 

.08375 

.09141 

37 

38 

.06415 

.06854 

.07044 

.07578 

.07701 

.08344 

.08387 

.09155 

38 

39 

.06425 

.06866 

.07055 

.07590 

.07713 

.08357 

.08399 

.09169 

39 

40 

.06435 

.06878 

.07065 

.07602 

.07724 

.08370 

.08410 

.09183 

40 

41 

.06445 

.06889 

.07076 

.07615 

.07735 

..08383 

.08422 

.09197 

41 

42 

.06456 

.06901 

.07087 

.07627 

.07746 

.08397 

.08434 

.09211 

42 

43 

.06466 

.06913 

.07098 

.07640 

.07757 

.08410 

.08445 

.09224 

43 

44 

.06476 

.06925 

.07108 

.07652 

.07769 

.08423 

.08457 

.09238 

44 

45 

.06486 

.06936 

.07119 

.07665 

.07780 

.08436 

.08469 

.09252 

45 

46 

.06497 

.06948 

.071:30 

.07677 

.07791 

.08449 

.08481 

.09266 

46 

47 

.06507 

.06960 

.07141 

.07690 

.07802 

.08463 

.08492 

.09280 

47 

48 

.06517 

.06972 

.07151 

.07702 

.07814 

.08476 

.08504 

.09294 

48 

40 

.06528 

.06984 

.07162 

.07715  ! 

.07825 

.08489 

.08516 

.09308 

49 

50 

.06538 

.06995 

.07173 

.07727 

.07836 

.08503 

.08528 

.09323 

50 

51 

.06548 

.07007 

.07184 

.07740 

.07848 

.08516 

.08539 

.09337 

51 

52 

.06559 

.07019 

.07195 

.07752 

07859 

.08529 

.08551 

.09351 

52 

53 

.06569 

.07031 

.07206 

.07765 

.07870 

.08542 

.08563 

.09365 

53 

54 

.06580 

.07043 

.07216 

.07778 

.07881 

.08556 

.08575 

.09379 

54 

55 

.06590 

.07055 

.07227 

.07790 

.07893 

.08569 

.08586 

.09393 

55 

56 

.06600 

.07067 

.07238 

.07803 

.07904 

.08582 

.08598 

.09407 

56 

57 

.06611 

.07079 

.07249 

.07816 

.07915 

.08596 

.08610 

.09421 

57 

58 

.06621 

.07091 

.07260 

.07828 

.07927 

.08609 

.08622 

.09435 

58 

59 

.00632 

.07103 

.07271 

.07841 

.07938 

.08623 

.08634 

.09449 

59 

60 

.06642 

.07115 

.07282 

.07853 

.07950 

.08636 

.08645 

.09464 

60 

TABLE  XXH.-VERSINES  AND  EXSEOANTS. 


' 

24° 

25°         26° 

27° 

/ 

Vers. 

Exsec. 

Vcrs. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.08645 

.094(54 

.09369 

.10338 

.10121 

.11200 

.10899 

.12233 

0 

1 

.08657 

.09478 

.09382 

.10353 

.10133 

.11276 

.10913 

.12249 

1 

2 

.08669 

.09492 

.09394 

.10368 

.10146 

.11292 

.10926 

.12266 

2 

3  i  .08681 

.09506 

.09406 

.10383 

.10159 

.11308 

.10939 

.12283 

3 

4 

.08693 

.09520 

.09418 

.10398 

.10172 

.11323 

.10952 

.12299 

4 

5 

.08705 

.09535 

.09431 

.10413 

.10184 

.11339 

.10965 

.12316 

5 

6 

.08717 

.09549 

.09443 

.10428 

.10197 

.11$55 

.1097'9 

.12333 

6 

7 

.08728 

.09563 

.09455 

.10443 

.10210 

.11371 

.10992 

.12349 

7 

8 

.08740 

.09577 

.09468 

.10458 

.10223 

.11387 

.11005 

.12366 

8 

9 

.08752 

.09592 

.09480 

.10473 

.10236 

.11403 

.11019 

.12383 

9 

10 

.08764 

.09606 

.09493 

.10488 

.10248 

.11419 

.11032 

.12400 

10 

11 

.08776 

.09620 

.09505 

.10503 

.10261 

.11435 

.11045 

.12416 

11 

12 

.08788 

.09635 

.09517 

.10518 

.10274 

.11451 

.11058 

.12433 

12 

13 

.08800 

.09649 

.09530 

.10533 

.10287 

.11467 

.11072 

.12450 

13 

14 

.08812 

.09663 

.09542 

.10549 

.10300 

.11483 

.11085 

..12467 

14 

15 

.08824 

.09678 

.09554 

.10564 

.10313 

.11499 

.11098 

.12484 

r> 

16 

.08*36 

.09692 

.09567 

.10579 

.10326 

.11515 

.11112 

.12501 

16 

17 

.08848  I  .09707 

.09579 

.10594 

.10338 

.11531 

.11125 

.12518 

17 

18 

.08860 

.09721 

.09592 

.10609 

.10351 

.11547 

.11138 

.12534 

18 

19 

.08873 

.09735 

.09604 

.10625 

.10364 

.11563 

.11152 

.12551 

19 

20 

.08884 

.09750 

.09617 

.10640 

.10377 

.11579 

.11165 

.12568 

20 

21 

.08896 

.09764 

.09629 

.10655 

.10390 

.11595 

.11178 

.12585 

21 

22 

.08908 

.09779 

.09642 

.10670 

.10403 

.11611 

.11192 

.12602 

22 

23 

.08920 

.09793 

.09654 

.10686 

.10416 

.11627 

.11205 

.12619 

23 

24 

.08932 

.09808 

.09666 

.10701 

.10429 

.11643 

.11218 

.12636 

24 

25 

.08944 

.09822 

.09679 

.10716 

.10442 

.11659 

.11232 

.12653 

25 

26 

.08956 

.09837 

.09691 

.10731 

.10455 

.11675 

.11245 

.12670 

26 

27 

.08968 

.09851 

.09704 

.10747 

.10468 

.11691 

.11259 

.12687 

27 

28 

.08980 

.09866 

.09716 

.10762 

.10481 

.11708 

.11272 

.12704 

28 

29 

.08992 

.09880 

.09729 

.10777 

.10494 

.11724 

.11285 

.12721 

29 

30 

.09004 

.09895 

.09741 

.10793 

.10507 

.11740 

.11299 

.12738 

30 

31 

.09016 

.09909 

.09754 

.10808 

.10520 

.11756 

.11312 

.12755 

31 

32 

.09028 

.09924 

.09767 

.10824 

.10533. 

.11772 

.11326 

.12772 

32 

33 

.09040 

.09939 

.09779 

.  10839 

.10546 

.11789 

.11339 

.12789 

33 

31 

.09052 

.09953 

.09792 

.10854 

.10559 

.11805 

.11353 

.12807 

34 

35 

.09064 

.09968 

.09804 

.10870   .10572 

.11821 

.11366 

.12824 

35 

36 

.09076 

.09982 

.09817 

.10885   .10585 

.11838 

.11380 

.12841 

36 

37 

.09089 

.09997 

.09829 

.10901 

.10598 

.11854 

.11393 

.12858 

37 

38 

.09101 

.10012 

.09842 

.10916 

.10611 

.11870 

.11407 

.12875 

38 

39 

.09113 

.10026 

.09854 

.10932 

.10024 

.11886 

.11420 

.12892 

39 

40 

.09125 

.10041 

.09867 

.10947 

.10637 

.11903 

.11434 

.12910 

40 

41 

.09137 

.10055 

.09880 

.10963 

.10650 

.11919 

.11447 

.12927 

41 

42 

.09149 

.10071 

.09892 

.10978 

.10663 

.11936 

.11461 

.12944 

42 

43 

.09161 

.10085 

.09905 

.10994 

.10676 

.11952 

.11474 

.12961 

43 

44 

.09174 

.10100 

.09918 

.11009 

.10689 

.11968 

.11488 

.12979 

44 

45 

.09186 

.10115 

.09930 

.11025 

.  10702 

.11985 

.11501 

.12996 

45 

46 

.09198 

.10130 

.09943 

.11041 

.10715 

.12001 

.11515 

.13013 

46 

47 

.09210 

.10144 

.09955 

.11056 

.10728 

.12018 

.11528 

.13031 

47 

48 

.09222 

.10159 

.09963 

.11072 

.10741 

.12034 

.11542 

.13048 

48 

49 

.09234 

.10174 

09981 

.  11087 

.10755 

.12051 

.11555 

.13065 

49 

50 

.09247 

.10189 

.09993 

.11103 

.10768 

.12067 

.11569 

.13083 

50 

51 

.09259 

.10204 

.10006 

.11119 

.10781 

.12084 

.11583 

.13100 

51 

52 

.09271 

.10218 

.10019 

.11134 

.10794 

.12100 

.11596 

.13117 

52 

53 

.09283 

.10233  ! 

.10032 

.11150 

.10807 

.12117 

.11610 

.13135 

53 

54 

.09296 

.10248 

.10044 

.11166 

.10820 

.12133 

.11623 

.13152 

54 

55 

.09308 

.10263 

.10057 

.11181 

.10833 

.12150 

.11637 

.13170 

55 

56 

.09320 

.10278 

.10070 

.11197 

.10847 

.12166 

.11651 

.13187 

56 

57 

.09&32 

.10293 

.10082 

.11213 

.10860 

.12183 

.11664 

.13205 

57 

58 

.09345 

.10308 

.10095 

.11229 

.10873 

.12199 

.11678 

.13222 

58 

59 

.09357 

.10323 

.10108 

.11244 

.10886 

.12216 

.11692 

.13240 

59 

60 

.09369 

.10338 

.10121 

.11260 

.10899 

.12233 

.11705 

.13257 

60 

vKi;siM-:s  AND  r.\sKr.\NTs. 


/ 

2 

8° 

2 

9° 

3( 

)• 

3 

L° 

t 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers.  ( 

Exsec. 

Vers. 

Exsec. 

~ 

0 

.11705 

.13257 

.12538 

.14335 

.13397 

.15470 

.14283 

.10603 

1 

.11719 

.13275 

.12552 

.14354 

.13412 

.15489 

.14298 

.10084 

1 

2 

.11733 

.13202 

.12566 

.  14372 

.13427 

.15509 

.14313 

.10704 

2 

8 

.11746 

.13310 

.12580 

.14391 

.13441 

.15528 

.14328 

.10725 

3 

4 

.11760 

.13327 

.12595 

.14409 

.13456 

.15548 

.14343 

.16745 

4 

5 

.11774 

.13345 

.12609 

.14428 

.13470 

.  15507 

.14358 

.16706 

5 

6 

.11787 

.13362 

.12623 

.14446 

.13485 

.15587 

.14373 

.16786 

6 

7 

.11801 

.13380 

.12637 

.14405 

.13499 

.15606 

.14388 

.10806 

7 

8 

.11815 

.13398 

.12651 

.14483 

.13514 

.15020 

.14403 

.10827 

8 

9 

.11828 

.13415 

.12665 

.14502 

.13529 

.15645 

.14418 

.16848 

9 

10 

.11*12 

.13433 

.12079 

.14521 

.13543 

.15005 

.14433 

.16868 

10 

11 

.11856 

.13451 

.12694 

.14539 

.13558 

.15684 

.14449 

.10889 

11 

12 

.11870 

.13468 

.12708 

.14558 

.13573 

.15704 

.14404 

.10909 

12 

13 

.11883 

.13486 

.12722 

.14576 

.13587 

.15724 

.14-179 

.10930 

13 

14 

.11897 

.13504 

.12736 

.14595 

.13602 

.15743 

.14-194 

.10950 

14 

15 

.11911 

.13521 

.12750 

.14614 

.13616 

.15763 

.14509 

.16971 

15 

16 

.11925 

.13539 

.12705 

.14632 

.13031 

.15782 

.14524 

.16992 

16 

17 

.11938 

.13557 

.12779 

.14651 

.13646 

.15802 

.14539 

.17012 

17 

18 

.11952 

.13575 

.12793 

.14670 

.13660 

.15822 

.14554 

.17033 

18 

19 

.11006 

.13593 

.12807 

.14689 

.13675 

.15841 

.14569 

.17C54 

19 

20 

.11980 

.13610 

.12822 

.14707 

.13690 

.15861 

.14584 

.17075 

20 

21 

.11994 

.13628 

.12836 

.14726 

.13705 

.15881 

.14599 

.17095 

21 

22 

.12007 

.13646 

.12850 

.14745 

.13719 

.15901 

.14615 

.17116 

22 

23 

.12021 

.13664 

.12864 

.14764 

.13734 

.15920 

.14630 

.17137 

23 

24 

.12035 

.13682 

.12879 

.14782 

.13749 

.15940 

.14645 

.  7158 

24 

25 

.12049 

.13700 

.12893 

.14801 

.137(33 

.15960 

.14660 

.  7178 

25 

26 

.12063 

.13718 

.12907 

.14820 

.13778 

.15980 

.14075 

.  7199 

26 

27 

.12077 

.13735 

.12921 

.14839 

.13793 

.16000 

.14090 

.  7220 

27 

28 

.12091 

.13753 

.12936 

.14858 

.13GC8 

.16019 

.14700 

.  7241 

28 

29 

.12104 

.13771 

.12950 

.14877 

.13822 

.16039 

.14721 

.  7362 

29 

30 

.12118 

.13789 

.12964 

.14896 

.13837 

.16059 

.14736 

.  7283 

30 

31 

.12132 

.13807 

.12979 

.14914 

.13852 

.16079 

.14751 

.17304 

31 

32 

.12146 

.13825 

.12993 

..14933 

.13867 

.16099 

.14766 

.17325 

32 

33 

.12160 

.13843 

.13007 

.1495° 

.13881 

.16119 

.14782 

.17346 

33 

34 

.12174 

.13861 

.13022 

.14971 

.13896 

.16139 

.14797 

.17367 

34 

35 

.12188 

.13879 

.13036 

.14990 

.13911 

.16159 

.14812 

.17388 

35 

36 

.12202 

.13897 

.13051 

.15009 

.13926 

.16179 

.14827 

.17409 

36 

37 

.12216 

.13916 

.13065 

.15028 

.13941 

.16199 

.14843 

.17430 

37 

38 

.12230 

.13934 

.13079 

.15047 

.13955 

.16219 

.14858 

.17451 

38 

39 

.12244 

.13952 

.13094 

.15006 

.13970 

.16239 

.14873 

.1747'2 

39 

40 

.12257 

.13970 

.13108 

.15085 

.13985 

.16259 

.14888 

.17493 

40 

41 

.12271 

.13988 

.13122 

.15105 

.14000 

.16279 

.14904 

.17514 

41 

42 

.12285 

.14006 

.13137 

.15124 

.14015 

.16299 

.14919 

.17535 

42 

43 

.12299 

.14024 

.13151 

.15143 

.14030 

.16319 

.14934 

.17556 

43 

44 

.12313 

.14042 

.13106 

.15102 

.14044 

.16339 

.14949 

.  7577 

44 

45 

.12327 

.14061 

.13180 

.15181 

.14059 

.16359 

.14965 

.17598 

45 

46 

.12341 

.14079 

.13195 

.15200 

.14074 

.16380 

.14980 

.17620 

46 

47 

.12355 

.14097 

.13209 

.15219 

.14089 

.16400 

.14995 

.17641 

47 

48 

.12369 

.14115 

.13223 

.15239 

.14104 

.16420 

.15011 

.17662 

48 

49 

.12383 

.14134 

.13238 

.15258 

.14119 

.16440 

.15026 

.17083 

49 

50 

.12397 

.14152 

.13252 

.15277 

.14134 

.16460 

.15041 

.17704 

50 

51 

.12411 

.14170 

.13267 

.15296 

.14149 

.16481 

.15057 

.17726 

51 

52 

.12425 

.14188 

.13281 

.15315 

.14164 

.16501 

.15072 

.17747 

52 

53 

.12439 

.14207 

.13296 

.15335 

.14179 

.16521 

.15087 

.17768 

53 

54 

.12454 

.14225 

.13310 

.15354 

.14194 

.16541 

.15103 

.17790 

54 

55 

.12468 

.14243 

.13325 

.15373 

.14208 

.16562 

.15118 

.17811 

55 

56 

.12482 

.14262 

.1.3339 

.15393 

.14223 

.16582 

.15134 

.17832 

56 

57 

.12496 

.14280 

.13354 

.15412 

.14238 

.16602 

.15149 

.17854 

57 

58 

.12510 

.14299 

.1,3368 

.15431 

.14253 

.16623 

.15164 

.17875 

58 

59 

.  12524 

.14317 

.13383 

.15451 

.14268 

.16643 

.15180 

.17896 

59 

60 

.12538 

.14335 

.13397 

.15470 

.14283 

.16663 

.15195 

.17918 

60 

TAKLK   XXII.    -VKKS1NKS  AND   EXSECANTS. 


' 

32° 

33C 

34° 

35° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.15195 

.17918 

.161:8 

.192MJ 

.17096 

.20622 

.18085 

.22077 

0 

1 

.15211 

.17939 

.10149 

.1-9259 

.17113 

.20645 

.18101 

.22102 

1 

2 

.15226 

.17961 

.10105 

.19281 

.17129 

.20669 

.18118 

.22127 

2 

3 

.  15241 

.17982 

.16181 

.19304 

.17145   .20693 

.18135 

.22152 

3 

4 

.15257 

.18004 

.16196 

.19327 

.17161   .20717 

.18152 

.22177 

4 

5 

.15272 

.18025 

.16212 

.19349 

.17178   .20740 

.18168 

.22202 

5 

6 

.15288 

.18047 

i  .16228 

.19372 

.17194   .20764 

.18185 

.22227 

6 

7   .15303 

.18068 

.16244 

.19394 

.17210   .20788 

1  .18202 

.22252 

7 

8 

.15319 

.18090 

.16260 

.19417 

.17227 

.20812 

.18218 

.22277 

8 

9 

.15334 

.18111 

.16276   .19440 

.17243 

.20836 

.18235 

.22302 

9 

10 

.15350 

.18133 

.16292   .19463 

.17259 

.20859 

.18252 

.22327 

10 

11 

.153G5 

.18155 

.16308   .19485 

.17276 

.20883 

.18269 

.22352 

11 

12 

.15381   .18176 

.16324   .19508 

.17292 

.20907 

.18286 

.22377 

12 

13 

.15396   .18198 

.16340   .19531 

.17308   .20931 

.18302 

.22402 

13 

14 

.15412   .18220 

.16355   .19554 

.17325 

.20955 

!l88*9 

.22428 

14 

15 

.15427  '  .18241 

.16371   .19576 

.17341 

.20979 

.18336 

.22453 

15 

16 

.15443   .18263 

.16387 

.19599 

.17357 

.21003 

.18353 

.22478 

16 

17 

.15458 

.18285 

.16403 

.19622 

.17374 

.21027 

.18369 

.22503 

17 

18 

.15474 

.18307 

.16419 

.19645 

.17390 

.21051 

.18386 

.22528 

18 

19 

.15489 

.18328 

.16435 

.19668 

.17407 

.21075 

.18403 

.22554  19 

20 

.15505 

.18350 

.16451 

.19691 

.17423 

.21099 

.18420 

.22579  20 

21 

.15520 

.18372 

.1G467 

.19713 

.17439 

.21123 

.18437 

.22604  21 

22 

.15536 

.18394 

.16483 

.19736 

.17456 

.21147 

.18454 

.22629 

22 

23 

.15552 

.18416 

.16499 

.19?'59 

.17472 

.21171 

.18470 

.22655 

23 

24 

.15567 

.18437 

.16515 

.19782   .17489   .21195 

.18487 

.22680 

24 

25 

.15583 

.18459 

.16531 

.19805  1  .17505   .21220 

.18504 

.22706 

25 

26 

.15598 

.18481 

.16547 

.19828 

.17522   .21244 

.18521   .22731 

26 

°7 

.15614 

.18503 

.16563 

.19851 

.17538   .21268 

.18538   .22756 

27 

28 

.15630 

.18525 

.16579 

.19874  :  .17554 

.21292 

.18555  1  .22782 

28 

29 

.15645 

.18547 

.16595 

.19897 

.17571 

.21316 

.18572  ;  .22807 

29 

30 

.15661 

.18569 

.16611 

.19920 

.17587 

.21341 

.18588 

.22833 

30 

31 

.15676 

.18591 

.16627 

.19944 

.17604 

.21365 

.18605 

.22858 

31 

32 

.15692 

.18613 

.16644 

.19967 

.17620 

.21389 

.18022 

.22884 

32 

33 

.15708 

.18635 

.16660  !  .19990 

.17637 

.21414 

.18639 

.22909 

33 

34 

.15723 

.18657 

.16676  i  .20013 

.17653   .21438 

.18656 

.229a5 

34 

35 

.15739 

.18679 

.16692   .20036 

.17670   .21462 

.18673 

.22960 

35 

36 

.15755   .18701 

.16708   .20059 

.17686  i  .21487 

.18690 

.22986 

36 

37 

.15170 

.18723 

.16724   .20083 

.17703 

.21511 

.18707 

.23012 

37 

38 

.15786 

.18745 

.16740   .20106 

.17719 

.21535 

.18724 

.23037 

38 

39 

.  15S02 

.18767 

.16756   .20129  : 

.17736 

.21560 

.18741 

.23063 

39 

40 

.15818 

.18790 

.16772 

.20152 

.17752 

.S1584 

.18758 

.23089 

40 

41 

.15833 

.18812 

.16788 

.20176 

.17769 

.21609 

.18775 

.23114 

41 

42 

.15849 

.18834 

.16805   .20199 

.17786 

.21633 

.18792 

.23140 

42 

43 

.15865 

.18856 

.16821   .20222 

.17802 

.21658 

.18809 

.23166 

43 

44 

.15880 

.18878 

.16837   .20246  j  .17819 

.21682 

.18826 

.23192 

44 

45 

.15896 

:  18901 

.16853  1  .20269 

.17835 

.21707 

.18843 

.23217 

45 

46 

.15912 

.18923 

.16869   .20292 

.17852 

.21731 

.18860 

.23243 

46 

47 

.15923 

.18945 

.16885   .20316 

.17868 

.21756 

.18877 

.23269 

47 

48 

.15943 

.18967 

.16902   .20339 

.17885 

.21781 

.18894 

.23295 

48 

49 

.15959 

.18990 

.16918 

.20363 

.17902 

.21805 

.18911 

.23321 

49 

50 

.15975 

.19012 

.16934 

.20386  ; 

.17918 

.21830 

.18928 

.23347 

50 

51 

.15991 

.19034 

.16950 

.20410 

.17935 

.2ia55 

.18945 

.23373 

51 

52 

.16006 

.19057 

.16966 

.20433 

.17952 

.21879 

.18962 

.23399 

52 

53 

.16022 

.19079 

.16983 

.20457   .17968 

.21904 

.18979 

.23424 

53 

54 

.10038 

.19102 

.16999 

.20480  i  .17985 

.21929 

.18996 

.23450 

54 

55 

.16054 

.19124   .17015 

.20504  1  .18001 

.21953 

.19013 

.23476 

55 

56 

.10070 

.19146 

.17031 

.20527  !'  .18018 

.21978 

.19030 

.23502 

56 

57 

!  16085 

.19169 

.17047 

.20551   .18035 

.22003 

.19047 

.23529 

57 

58 

.16101 

.19191 

.17064 

.205;-)   .18051 

.22028 

.19064 

.23555 

58 

59 

.16117 

.19214 

.17080 

.20598 

.18008 

.22053 

.19081 

.23581 

59 

60 

.16133 

.19236   .17096 

.20022   .18085 

.22077  U  .19098 

.23607 

60 

36S 


AULE  xxii.— VEfcsiNEs  AND  EXSECANTS. 


/ 

36° 

37° 

38° 

30° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers.  Exsec. 

~T 

.19098 

.23007 

.201S6 

.25214 

.21199 

.26902 

.22285 

.28676 

0 

i 

.19115 

.23633 

.20154 

.25241 

.21217 

.26931 

.22304 

.287'06 

1 

2 

.19133 

.23659 

.20171 

.25269 

.21235 

.26960 

.22322 

.28737 

2 

3 

.19150 

.23685 

.20189 

.25296 

.21253 

.26988 

.22340 

.28767 

3 

4 

.19107 

.23711 

.20207 

.25324 

.21271 

.27017 

.22359 

.28797 

4 

5 

.1918! 

.23738 

.20224 

.25351 

.21289 

.27046 

.22377 

.28828 

5 

6 

.19201 

.23764 

.20242 

.25379 

.21307 

.27075 

.22395 

.28858 

6 

7 

.19218 

.23790 

.20259 

.25406 

.21324 

.27104 

.22414 

.28889 

7 

8 

.19235 

.23816 

.20277 

.25434 

.21342 

.27133 

.22432 

.28919 

8 

9 

.19252 

.23843 

.20294 

.25462 

.21360 

.27162 

.22450 

.28950 

9 

10 

.19270 

.23869 

.20313 

.25489 

.21378 

.27191 

.22469 

.28980 

10 

11 

.19287 

.23895 

.20329 

.25517 

.21396 

.27221 

.22487 

.29011 

11 

12 

.19304 

.23922 

.20347 

.25545 

.21414 

.27250 

.22500 

.29042 

12 

13 

.19321 

.23948 

.20365 

.25572 

.21432 

.27279 

.22524 

.29072 

13 

u 

.19338 

.23975 

.20382 

.25600 

.21450 

.27308 

.22542 

.29103 

14 

15 

.19356 

.24001 

.20400 

.25628 

.21468 

.27337 

.22561 

.29133 

15 

16 

.19373 

.24028 

.20417 

.25656 

.21486 

.27366 

.22579 

.29164 

16 

17 

.19390 

.24054 

.20435 

.25683 

.21504 

.27396 

.22598 

.29195  !  17 

18 

.19407 

.24081 

.20453 

.25711 

.21522 

.27425 

.22616 

.29226 

18 

19 

.19424 

.24107 

.20470 

.25739 

.21540 

.27454 

.22634 

.29256 

19. 

20 

.19442 

.24134 

.20488 

.25767 

.21558 

.27483 

.22653 

.29287 

20 

21 

.19459 

.24160 

.20506 

.25795  i 

.21576 

.27513 

.22671 

.29318 

21 

22 

.19476 

.24187 

.20523 

.25823  i 

.21595 

.27542 

.22690 

.29349 

22 

23 

.19493 

.24213 

.20541 

.25851  i 

.21613 

.27572 

.22708 

.29380 

23 

24 

.19511 

.24240 

.20559 

.25879  j 

.21631 

.27601 

.22727 

.29411 

24 

25 

.19528 

.24267 

.20576 

.25907  i 

.21649 

.27630 

.22745 

.29442 

25 

26 

.19545 

.24293 

.20594 

.25935 

.21667 

.27660 

.22764 

.29473 

26 

27 

.19502 

.24320 

.20612 

.25963 

.21685 

.27689 

.22782 

.29504 

27 

28 

.19580 

.24347 

.20629 

.25991 

.21703 

.27719 

.22801 

.29535 

28 

29 

.19597 

.24373 

.20647 

.26019 

.21721 

.27748 

.22819 

.29566 

29 

30 

.19614 

.24400 

.20665 

.26047 

.21739 

.27778 

.22838 

.29597 

30 

31 

.19632 

.24427 

.20682 

.26075 

.21757 

.27807 

.22856 

.29628 

31 

82 

.19649 

.24454 

.20700 

.26104 

.21775 

.27837 

.22875 

.29659 

32 

33 

.19666 

.24481 

.20718 

.26132 

.21794 

.27867 

.22893 

.29690 

33 

34 

.1^684 

.24508 

.20736 

.26160 

.21812 

.27896 

.22912 

.29721 

34 

35 

.19701 

.24534 

.20753 

.26188 

.21830 

.27926 

.22930 

.29752 

35 

36 

.19718 

.24561 

.20771 

.26216 

.21848 

.27956 

.22949 

.29784 

36 

37 

.19736 

.24588 

.20789 

.26245 

.21866 

.27985 

.22967 

.29815 

37 

38 

.19753 

.24615 

.20807 

.26273 

.21884 

.28015 

.22986 

.29846 

38 

39 

.19770 

.21642 

.20824 

.26301 

.21902 

.28045 

.23004 

.29877 

39 

40 

.19788 

.24669 

.20842 

.26330 

.21921 

.28075 

.23023 

.29909 

40 

41 

.19805 

.24696 

.20860 

.26358 

.21939 

.28105 

.23041 

.29940 

41 

42 

.19822 

.24723 

.20878 

.26387 

.21957 

.28134 

.23060 

.29971 

42 

43 

.19840 

.24750 

.20895 

.26415 

.21975 

.28164 

.23079 

.30003 

43 

44 

.19857 

.24777 

.20913 

.26443 

.21993 

.28194 

.23097 

.30034 

44 

45 

.19875 

.24804 

.20931 

.26472 

.22012 

.28224 

.23116 

.30066 

45 

46 

.19892 

.24832 

.20949 

.26500 

.22030 

.28254 

.23134 

.30097 

48 

47 

.19909 

.24859 

.20967 

.26529 

.22048 

.28284 

.23153 

.30129 

47 

48 

.19927 

.24886 

.20985 

.26557 

.22066 

.28314 

.23172 

.30160 

43 

49 

.19944 

.24913 

.21002 

.26586 

.22084 

.28344 

.23190 

.30192 

49 

50 

.19962 

.24940 

.21020 

.26615 

.22103 

.28374 

.23209 

.30223 

50 

51 

.19979 

.24967 

.21038 

.26643 

.22121 

.28404 

.23228 

.30255 

61 

r  Q 

.19997 

.24995 

.21056 

.26072 

.22139 

.28434 

.23246 

.30287 

52 

53 

.20014 

.25022 

.21074 

.26701 

.22157 

.28464 

.23265 

.30318 

53 

54 

.20032 

.25049 

.21092 

.26729 

.22176 

.28495 

.23283 

.30350 

54 

55 

.20049 

.25077 

.21109 

.26758 

.22194 

.28525 

.23302 

.30382 

55 

56 

.20066 

.25104 

.21127 

.26787 

.22212 

.28555 

.23321 

.30413 

56 

57 

.900*4 

.25131 

.21145 

.26815 

.22231 

.28585 

.23339 

.30445 

57 

58 

.20101 

.25159 

.21163 

.26844 

.22249 

.28615 

.23358 

.30477 

53 

59 

.20119 

.25186 

.21181 

.26873 

.22267 

.28646  i 

.23377 

.30509 

59 

60 

.20136 

.25214 

.21199 

.26902  1 

.22285 

.28676 

.23396 

.30541 

60 

TABLE  XXII.— VERSINES  AND  EXSECANTS. 


369 


' 

40° 

41° 

42° 

43° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.23396 

.30541 

.24529- 

.32501 

.25686 

.34563 

.26865 

.36733 

0 

1 

.23414 

.30573 

.24548 

.32535 

.25705 

.34599 

.26884 

.36770 

1 

2 

.23433 

.30605 

.24567 

.32568 

.25724 

.34634 

.26904 

.36807 

2 

3 

.23452 

.30636 

.24586 

.32602 

.25744 

.34669 

.26924 

.36844 

3 

4 

.23470 

.30668 

.24605 

.32636 

.25763 

.34704 

.26944 

.36881 

4 

5 

.23489 

.30700 

.24625 

.32669 

.25783 

.34740 

.26964 

.36919 

5 

6 

.23508 

.30732 

.24644 

.32703 

.25802 

.34775 

.26984 

.36956 

6 

7 

.23527 

.30764 

.24663 

.32737 

.25822 

.34811 

.27004 

.36993 

7 

8 

.23545 

.30796 

.24682 

.32770 

.25841 

.34846 

.27024 

.37030 

8 

9 

.23564 

.30829 

.24701 

.32804 

.25861 

.34882 

.27043 

.37068 

9 

10 

.23583 

.30861 

.24720 

.32838 

.25880 

.34917 

.27063 

.37105 

10 

11 

.23603 

.30893 

.24739 

.32872 

.25900 

.34953 

.27083 

.37143 

11 

12 

.23620 

.30925 

.24759 

.32905 

.25930 

.34988 

.27103 

.37180 

12 

13 

.23639 

.30957 

.24778 

.32939 

.25939 

.35024 

.27123 

.37218 

13 

14 

.23658 

.30989 

.24797 

.32973 

.25959 

.35060 

.27143 

.37255 

14 

15 

.23677 

.31022 

.24816 

.33007 

.25978 

.35095 

.27163 

.37293 

15 

16 

.23696 

.31054 

.24835 

.33041 

.25998 

.35131 

.27183 

.37330 

16 

17 

.23714 

.31086 

.24854 

.33075 

.26017 

.35167 

.27203 

.37368 

17 

18 

.23733 

.31119 

.24874 

.33109 

.26037 

.35203 

.27223 

.37406 

18 

19 

.23752 

.31151 

.24893 

.33143 

.26056 

.35238 

.27243 

.37443 

19 

20 

.23771 

.31183 

.24912 

.33177 

.26076 

.35274 

.27263 

.37481 

20 

21 

.23790 

.31216 

.24931 

.33211 

.26096 

.35310 

.27283 

.37519 

21 

22 

.23808 

.31248 

.24950 

.33245 

.26115 

.35346 

.27303 

.37556 

22 

23 

.23827 

.31281 

.24970 

.33279 

.26135 

.35382 

.27323 

.37594 

23 

24 

.23846 

.31313 

.24989 

.33314 

.26154 

.35418 

.27343 

.37632 

24 

25 

.23865 

.31346 

.25008 

.33348 

.26174 

.35454 

.27363 

.37670 

25 

26 

.23884 

.31378 

.25027 

.33382 

.26194 

.35490 

.27383 

.37708 

26 

27 

.23903 

.31411 

.25047 

.33J16 

.26213 

.35526 

.27403 

.37746 

27 

28 

.23922 

.31443 

.25066 

.33451 

.26233 

.35562 

.27423 

.37784 

28 

29 

.23941 

.31476 

.25085 

.33485 

.26253 

.35598 

.27443 

.37822 

29 

30 

.23959 

.31509 

.25104 

.33519 

.26272 

.35634 

.27463 

.37860 

30 

31 

.23978 

.31541 

.25124 

.33554 

.26292 

.35670 

.27483 

.37898 

31 

32 

.23997 

.31574 

.25143 

.33588 

.26312 

.35707 

.27503 

.37936 

32 

33 

.24016 

.31607 

.25162 

.33622 

.26331 

.35743 

.27523 

.37974 

33 

34 

.24035 

.31640 

.25182 

.33657 

.26351 

.35779 

.27543 

.38012 

34 

35 

.24054 

.31672 

.25201 

.33691 

.26371 

.35815 

.27563 

.38051 

35 

36 

.24073 

.31705 

.25220 

.33726 

.26390 

.35852 

.27583 

.38089 

36 

37 

.24092 

.31738 

.25240 

.33760 

.26410 

.35888 

.27603 

.38127 

37 

38 

.24111 

.31771 

.25259 

.33795 

.26430 

.35924 

.27623 

.38165 

33 

39 

.24130 

.31804 

.25278 

.33830 

.26449 

.35961 

.27643 

.38204 

39 

40 

.24149 

.31837 

.25297 

.33864 

.26469 

.35997 

.27663 

.38242 

40 

41 

.24168 

.31870 

.25317 

.33899 

.26489 

.36034 

.27683 

.38280 

41 

42 

.24187 

.31903 

.25336 

.33934 

.26509 

.36070 

.27703 

.38319 

4:? 

43 

.24206 

.31936 

.25356 

.33968 

.26528 

.36107 

.27723 

.38357 

43 

44 

.24225 

.31969 

.25375 

.34003 

.26548 

.36143 

.27743 

.38396 

4i 

45 

.24244 

.32002 

.25394 

.34038 

.26568 

.36180 

.27764 

.38434 

43 

46 

.24262 

.32035 

.25414 

.34073 

.26588 

.36217 

.27784 

.38473 

46 

47 

.24281 

.32068 

.25433 

.34108 

.26607 

.36253 

.27804 

.38512 

47 

48 

.24300 

.32101 

.25452 

.34142 

.26627 

.36290 

.27824 

.38550 

43 

49 

.24320 

.32134 

.25472 

.34177 

.26647 

.36327 

.27844 

.38589 

43 

50 

.24339 

.32168 

.25491 

.34212 

.26667 

.36363 

.27864 

.38628 

50 

51 

.24358 

.32201 

.25511 

.34247 

.26686 

.36400 

.27884 

.38666 

51 

52 

.24377 

.32234 

.25530 

.34282 

.26706 

.36437 

.27905 

.38705 

5:3 

53 

.24396 

.32267 

.25549 

.34317 

.26726 

.36474 

.27925 

.38744 

53 

54 

.24415 

.32301 

.25569 

.34352 

.26746 

.36511 

.27945 

.38783 

54 

55 

.24434 

.32334 

.25588 

.34387 

.26766 

.36548 

.27965 

.38822 

55 

56 

.24453 

.32368 

.25608 

.34423 

.  .26785 

.36585 

.27985 

.38860 

56 

57 

.24472 

.32401 

.25627 

.34458 

.26805 

.36622 

.28005 

.38899 

57 

58 

.24491 

.32434 

.25647 

.34493 

.26825 

.36659 

.28026 

.38938 

58 

59 

.24510 

.32468 

.25666 

.34528 

.26845 

.36696 

.28046 

.38977 

59 

60 

.24529 

.32501 

.25686 

.34563 

.26865 

.ijG733  | 

.28066 

.39016 

60 

TAHLK.  \\n.    VKKSINKS  ANP  KXSI-VANTS. 


44° 

45°         46°         47° 

Vers. 

BXSBO. 

V(M-S. 

Kxstv. 

\Vrss.  Kxscv.   Vrrs.  Kxser. 

' 

0 

.88066 

TsTwiT 

.29289 

.41121 

.30584 

.•*;'.).'.(•>   .31SOO   .411628 

0 

1 

.88088 

.89065 

.29310 

.41468 

.80555 

.1:  !'.);)<)   .:us-ji 

.40674 

1 

9 

.28108 

.89095 

.29830 

.41604 

.80576 

.44042 

.31848 

.46719 

2 

3 

38127 

.89184 

.29851 

.41545 

.80597 

.44086 

.81864 

.40705 

8 

4 

[28147 

.39173 

[29372 

.41586 

.80818 

.  Hi2!) 

.48811 

4 

5 

.38167 

.39912 

.29398 

.m>27   .3oti;«) 

.44178   .:M'.H>7 

.40857 

5 

G 

.28187 

.39251 

[29418 

.41669 

.80660 

.11217   .:»!»28 

.46903 

6 

7 

.28808 

.89991 

.29433 

.41710 

.30681 

.41260   .:51!H!) 

.46949 

7 

8 

.  2S22S 

.39330 

.29454 

.41759 

.80708 

.44804 

.31971 

.46995 

8 

0 

.  2S2  1  S 

[41793 

.80728 

.44347 

.81992 

.•17011 

9 

10 

.38988 

.3940S) 

.99495 

.41835 

.30744 

.44391 

.32013 

.47067 

10 

It 

,28289 

.89448 

.90516 

.41876 

.30765 

.44485 

.89085 

.47184 

11 

u 

.28309 

39487 

.•ir.u  -; 

.44479 

.82056 

.47180 

12 

13 

.28329 

.39527 

.41959 

[30807 

.44523 

.32077 

[47226 

13 

14 

[39568 

.29578 

.42001 

[80828 

.44567 

.32099 

.-17272 

14 

15 

.89606 

.29599 

.10012 

.80849 

.44610 

.32120 

.47819 

15 

13 

.28890 

.89646 

.29619 

.42084 

.30870 

.44654 

.32141 

.47865 

10 

17 

[98410 

.39685 

.29640 

.48126 

.80891 

.44898 

.88163 

.47411 

17 

18 

.26431 

.89601 

.421G8 

.80912 

.44742 

.82184 

.47458 

18 

19 

.28461 

.89764 

.29081 

[42210 

.44787 

.82205 

.47604 

19 

80 

[98471 

[89804 

.49951 

.30951 

.44881 

.82287 

.41551 

20 

81 

.28499 

.89844 

.99793 

.30075 

.44875 

.32948 

.47598 

21 

.28518 

.89884 

.42335 

.44919 

.83270 

.47644 

22 

83 

.28582 

.39924 

.29764 

.42377 

.31017 

.44963 

.47691 

83 

XT  t 

.28553 

.39963 

.29785 

.42419 

.81038 

.45007 

.32812 

.47788 

N 

85 

.28573 

[40008 

.48461 

.81059 

.45052 

.32384 

.47784 

95 

88 

.40043 

.29826 

.42508 

.81080 

.45096 

.4783] 

86 

97 

.'.'SOI  ( 

.40083 

.29847 

.49545 

.81101 

.45141 

.89377 

.47878 

27 

88 

.28634 

.40198 

.49587 

.81122 

.45185 

.82398 

.47925 

2S 

99 

.28655 

.40168 

.29888 

.42680 

.81143 

.47972 

99 

GO 

[98875 

.40908 

.90909 

.42072 

.31105 

,45874 

.32111 

.48019 

30 

si 

.98695 

.IOC  13 

.29030 

.1271  I 

.31180 

.45819 

.89482 

.48086 

31 

82 

.88716 

.40283 

.99951 

.  12756 

.81207 

.45363 

.32484 

.48113 

8S 

88 

[28788 

.40324 

.99971 

.49799 

.81228 

.45408 

.48160 

33 

34 

.88757 

.40364 

.29992 

.42841 

.31219 

.45452 

.82527 

.48207 

84 

85 

[28777 

,40404 

.30013 

.49888 

.81270 

.45497 

.32548 

.  18254 

;tr> 

36 

.28797 

.40  III 

.30034 

.42926 

.81291 

.45542 

.82570 

.48801 

30 

87 

.28818 

.40485 

.88054 

.49968 

.81812 

.45587 

.82591 

.48849 

37 

38 

.40595 

.30075 

.43011 

.81884 

.45681 

.82813 

.48398 

88 

39 

.88859 

.40565 

.30096 

.48058 

.81855 

.45678 

.32884 

.48448 

39 

-10 

.40000 

.30117 

.48098 

.31370 

,45721 

.89668 

.48491 

40 

41 

.28900 

.40646 

.30138 

.43139 

.31307 

.45766 

.89677 

.48588 

41 

12 

.30158 

.48181 

[31418 

.45811 

.32899 

.48586 

42 

43 

.88941 

.40797 

.80179 

.43224 

.81489 

.45856 

.32720 

.48633 

43 

44 

.28901 

.40708 

.80200 

.48867 

.81461 

.45901 

.327:2 

.48681 

44 

45 

.28981 

.40808 

.80221 

.43810 

.81482 

.45946 

.32763 

.48728 

45 

46 

.29008 

.40849 

.80942 

.43352 

.81508 

.45992 

.82785 

.48776 

46 

47 

.29028 

.40890 

.30203 

.48895 

.81584 

.46037 

.82806 

.48824 

47 

48 

[89048 

.40930 

.30283 

.43488 

.81545 

.46062 

.38828 

.48871 

48 

49 

.89068 

.40971 

.30304 

.43481 

.81587 

.-11  H27 

.82849 

.48919 

49 

50 

.29084 

.41012 

.80395 

.43524 

.31588 

.40173 

.88871 

.4SU67  50 

51 

.29104 

.41053 

.30346 

.43567 

.31009 

.46218 

.32S93 

.49015 

51 

52 

.29125 

.41093 

.30307 

.43010 

.81030 

..(('-•.Hi;? 

.."2;M  1 

.49063 

58 

53 

.80145 

.41134 

.30388 

.43053 

.31051 

.48309 

.82986 

.49111 

53 

54 

.89166 

.41175 

.30409 

.48696 

.31073 

.46354 

[82957 

.49159 

64 

55 

.89187 

.41216 

.30430 

.48789 

.31694 

.46400 

.321)79 

.49207 

55 

56 

.89907 

.41257 

.80451 

.43783 

.31715 

.46445 

.88001 

56 

57 

89928 

.41298 

.30471 

.48826 

.31730 

.46491 

.83022 

[49808 

57 

58 

.99948 

.41839 

[30492 

.43869 

.81758 

.88044 

58 

59 

.99989 

.41380 

.80518 

.43912 

.31779 

.46582 

.49399 

59 

60 

.90989 

.41491 

.30534 

.48956 

.31800 

.46628 

.33087 

.49443 

60 

TAKLK   XXII.— VERSINES   AND    KXSErANTS 


t 

48° 

49° 

50° 

51° 

f 

Vers. 

Exsec. 

Vers. 

E::sec. 

Vers. 

Exsec. 

Vers. 

Kxstx-. 

o  .33087 

.49448   .34304 

.52425 

.35721 

.55572 

.37068 

.58902 

0 

1   .33109 

.49496 

.34410 

.52471) 

.35744 

.55626   .37091 

.58968 

1 

2   .33130 

.49544  : 

.84438 

.58687 

.35766 

.55080   .37113   .59016 

2 

3   .33152 

.34460 

.52579 

.35788 

.55734   .37130   .59073 

3 

4   .33173 

.49641   .34482 

.59680 

.35810 

.55789  1  .37158 

.59130 

4 

5   .33195 

.49090   .34504 

.52681 

.35833 

.55843 

.37181 

.59188 

5 

6   .33217 

.49738 

.84536 

.52732 

.35855 

.55897 

.37204   .59245   6 

7   .3323$ 

.49787 

.34548 

.52784 

.35877 

.55951 

.372215   .59300 

7 

8   .33260 

.49835 

.34570 

.35900 

.56005 

.37249 

.59360 

8 

9 

.33282 

.49884 

.34592 

.52886 

.85922 

.56060 

.37272 

.59418 

9 

10 

.33303 

.49933 

.34614 

.52938 

.35944 

.56114 

.87894 

.59475 

10 

11 

.33325 

.49981 

.34636 

.52989 

.35967 

.56169 

.37317 

.59533 

11 

12   .33347 

.50030 

.34658 

[53041 

.35989 

.56833 

.37340 

.59590 

12 

13   .33303 

.50079 

.346:X> 

.53092 

.36011 

.37362 

.59648 

13 

14 

.33390 

.501:28 

.34702 

.53144 

.36034 

.37385 

.59706 

14 

13 

.33412 

.50177 

.31721 

.53196 

.36056 

.56387 

.37408 

.59764 

15 

HI 

.33434 

[50236 

.34746 

.53247 

.36078 

.56442 

.37430 

.69823 

10 

17 

.33455 

.50275 

.34768 

.53299 

.36101 

.56497   .  37453 

.59880 

17 

18 

.33477 

.50324 

.34790 

.53351 

.36123 

.56551 

i  .37476 

.59938 

13 

19 

.33199 

.50373 

.31812 

.5:1403   .36146 

.56606 

.37498 

.59996 

19 

20 

.33520 

.50422 

.34834 

.53455 

.36168 

.56661 

.37521 

.60054 

30 

21 

.33542 

.50471 

.34856 

.53507 

.36190 

.56716 

.37544 

.60112 

21 

22 

.33504 

.50521  |  .34878 

.53559 

.36213 

.50771 

.37567 

.60171 

22 

23 

.33580 

.5(1570  : 

.34900 

.53611 

.36235 

.37589 

23 

<H 

.33007 

.50619 

.34923 

.53663 

.36258 

.56881 

.37612 

.60287 

24 

25 

.33629 

.50869 

.34945 

.53715 

.36280 

.56937 

.37635 

.6«3346 

25 

26 

.33651 

.50718 

.34967 

.53768 

.36302 

.56992 

.37658 

.60404 

26 

'T 

.33673 

.50767 

.34989 

.53820 

.36325 

.57047 

.37680 

.60463 

27 

28 

.33694 

.60817 

.35011 

.53872 

.36347 

.57103 

.37703 

.60521 

28 

29 

.33716 

.50866 

.35033 

.53924 

.36370 

.57158 

.37726 

.60580 

29 

30 

.33738 

.50916 

.35055 

.53977 

.36393 

.57213 

.37749 

.60639 

30 

31 

.33760 

.50966 

.35077 

.54029 

.36415 

.57269 

.37771 

.60698 

31 

32   .33782 

.51015 

.35099 

.54082 

.36437 

.57324 

.37794 

.60756 

32 

3.°>   .33S03 

.51065 

.35122 

.54134 

.36460 

.57380 

.37817 

.60815 

33 

34   .33S25 

.51115 

.35144 

.54187 

.36482 

.57436 

.37840 

.60874 

34 

35   .33847 

.51165 

.35106 

.54240 

.36501 

.57491 

.37862 

.60933 

35 

36   .33809 

.51215 

'.35188 

.54292 

.36527 

.57547 

[87885 

.60992 

36 

37   .33891 

.51265 

.35210 

.54345 

.36549 

.57603 

.37908 

.61051 

37 

38   .33912 

.51314 

.35232 

.54398 

.3657'2 

.57659 

.37931 

.61111 

38 

39   .33934 

.51364 

.35254 

.54451 

.36594 

.57715 

.37951 

.61170 

39 

40   .33956 

.51415 

.35277 

.54504 

.36617 

.57771 

.37976 

.61229 

40 

41  I  .33978 

.51465 

.35299 

.54557 

.36639 

.57827  i  .379P9 

.61283 

41 

42   .34000 

.51515 

.35321 

.54610 

.36602 

.57883  1  .38022 

.61348 

42 

43  !  .34022 

.51565 

.35343 

.54663 

.36684 

.57939  ||  .38045 

.61407 

43 

44 

.34044 

.51615 

.35365 

.54716 

.36707 

.57995   .38068 

.61467 

44 

45 

.34065 

.51665 

.85388 

.54769 

.36729 

.58051  !  .38091 

.61526 

45 

46 

.34087 

.51716 

.35410 

.54822 

.36752 

.58108 

.38113 

.61586 

46 

47 

.34109 

.51766 

.85433 

.54876 

.36775 

.58164 

.38136 

.61646 

47 

48 

.34131 

.51817 

.35454 

.54929 

.36797 

.5-001 

.38159 

.61705 

48 

49 

.34153 

.51867 

.£5476 

.54982 

.36820 

[58277 

[88188 

.61765 

49 

50 

.34175 

.51918 

.35499 

.55036 

.36842 

.58333 

.38205 

.61825 

50 

51 

.34197 

.51968 

.35521 

.55089 

.36865 

.58390 

.38228 

.61885 

51 

52 

.34219 

.52019 

.35543 

.55143 

.86887 

.58447 

.38251 

.61945 

52 

53 

.34241 

[69069 

.35565 

.55196 

.36910 

.58503 

.38274 

.62005 

53 

54 

.34262 

.52120 

.35588 

.55250 

.36932 

.58560 

[88896 

.62065 

54 

55  !  .31284 

.52171 

.35610 

.55303 

.36955 

.58617 

.38319 

.62125 

55 

56   .34306 

.522-22 

.35632 

.55357 

.36978 

.58674 

.38342 

.62185 

56 

57 

.34328 

.52273 

.35654 

.55411 

.37000 

.58731 

.38365 

.62246 

57 

53 

.34350 

.52323 

.35877 

.5541.5   .37023 

.58788 

.88888 

.62306 

58 

59   .34372 

.52374 

.35699 

.37045 

.58845 

.38411 

.62366 

59 

60   .34394 

.52425 

.35721 

.55572   .37008 

.58903 

.38434 

.62427 

6Q 

TABLE   XXII.     YERSINES  AND   EXSEL'ANTS. 


52° 

53° 

54° 

55° 

YITS.  Exsec. 

Vers.  Exsec. 

Vers. 

Exsec.   Vers. 

Exsec. 

0 

.38434 

.62427 

.39819 

.06164  |  .41221 

.70130 

.42642 

.74345 

0 

1 

.38457 

.62487 

.39842 

.66228 

.41245 

.70198 

.42666 

.74417 

1 

o 

.38480 

.62548  ! 

.39865 

.66292 

.41269 

.702(57 

.42690 

.74490 

2 

3 

.38503 

.62609 

.39888 

.66357 

.41292 

.70335   .42714 

.74568 

3 

4 

.38526 

.62669 

.39911 

.66421 

.41316 

.70403   .42738 

.74635 

4 

5 

.38549 

.02730 

.39935 

.66486 

.41339 

.70472   .42702 

.74708 

5 

6 

.38571 

.(52791 

.39958 

.66550 

.41363 

.70540  ll  .42785 

.74781 

6 

7 

.38594 

.62852  : 

.39981 

.66615 

.41386 

.70809 

.42809 

.74854 

7 

8 

.38617 

.62913  ! 

.40005 

.66679 

.41410 

.70677  i 

.42833 

.74927 

8 

9 

.38640 

.62974  ' 

.40028 

.66744 

.41433 

.70746  i 

.42857 

.75000 

9 

10 

.38663 

.63035 

.40051 

.66809 

.41457 

.70815  ; 

.42881 

.75073 

10 

11 

.38686 

.63096 

.40074 

.66873 

.41481 

.70884 

.42905 

.75146 

11 

12 

.38709 

.63157 

.40098 

.66938 

.41504 

.70953   .42929 

.75219   12 

13 

.38732 

.63218 

.40121 

.67003 

.41528 

.71022  !  .42953 

.75293  !  13 

14 

.38755 

.63279  j 

.40144 

.67068 

.41551 

.71091   .42976   .75366  !  14 

15 

.38778 

.63341  i 

.40168 

.67133 

.41575 

.71160 

.43000   .75440  15 

16 

.38801 

.63402 

.40191 

.67199 

.41599 

.71229 

.43024   .75513  16 

17 

.38824 

.63464 

.40214 

.67264 

.41622 

.71298 

.43048   .75587  !  17 

18 

.38847 

.63525 

.40237 

.67329 

.41646 

.71368  i 

.43072   .75661   18 

19 

.38870 

.63587  ! 

.40261 

.67394 

.41670 

.71437 

.43096 

.75734  19 

20 

.38893 

.63&4S  ; 

.40284 

.67460 

.41693 

.71506  : 

.43120 

.75808  20 

21 

.38916 

.63710 

.40307 

.67525 

.41717 

.71576 

.43144 

.75882 

21 

22 

.38939 

.63772 

.40331 

.67591 

.41740 

.71646 

.43168 

.75956 

22 

23 

.38902 

.63834 

.40354 

.67656 

.41764 

.71715 

.43192 

.76031 

23 

24 

.38985 

.63895 

.40378 

.67722 

.41788 

.71785 

.43216 

.76105 

24 

25 

.39009 

.63957 

.40401 

.67788 

.41811 

.71855 

.43240 

.76179 

25 

26 

.39032 

.64019 

.40424 

.67853 

.41835 

.71925 

.43264 

.76253 

26 

27 

.39055 

.64081 

.40448 

.67919 

.41859 

.71995  I 

.43287 

.76328 

27 

28 

.39078 

.64144 

.40471 

.67985 

.41882 

.72065 

.43311 

.76402 

28 

29 

.39101 

.64206 

.40494 

.68051 

.41906 

.72135 

.43335 

.76477 

29 

30 

.39124 

.64268 

.40518 

.68117 

.41930 

.72205 

.43359 

.  76552 

30 

31 

.39147 

.64330 

.40541 

.68183 

.41953 

.72275 

.43383 

.76626 

31 

32 

.39170 

.64393 

.40565 

.68250 

.41977 

.72346 

.43407 

.76701 

32 

33 

.39193 

.64455 

.40588 

.68316 

.42001 

.72416 

.43431 

.76776 

33 

34 

.39216 

.04518 

.40611 

.68382 

.42024 

.72487 

.43455 

.76851 

34 

35 

.39239 

.64580 

.40635 

.68449 

.42048 

.72557 

.43479 

.76926 

35 

36 

.39262 

.64643 

.40658 

.68515 

.42072 

.72628 

.43503 

.77001 

36 

37 

.39286 

.64705 

.40682 

.68582 

.42096 

.72698 

.43527 

.77077 

37 

38 

.39309 

.64768 

.40705 

.68648 

.42119 

.72769  ! 

.43551 

.77152 

38 

39 

.39332 

.64831 

.40728 

.68715 

.42143 

.72840 

.43575 

.77227 

39 

40 

.39355 

.64894 

.40752 

.68782 

.42167 

.72911 

.43599 

.77303 

40 

41 

.39378 

.64957 

.40775 

.68848 

.42191 

.72982  ' 

.43623 

.77378 

41 

42 

.39401 

.65020 

.40799 

.68915 

.42214 

.73053 

.43647 

.77454 

42 

43 

.39424 

.65083 

.40822 

.68982 

.42238 

.73124 

.43671 

.77530 

43 

44 

.39447 

.65146 

.40846 

.69049 

.42262 

.73195 

.43695 

.77606 

44 

45 

.39471 

.65209 

.40869 

.69116 

.42285 

.73267 

.43720 

.77681 

45 

46 

.39494 

.65272 

.40893 

.69183 

.42309 

.73338  j 

.43744 

.77757 

46 

47 

.39517 

.65336 

.40916 

.69250 

.42333 

.73409 

.43768 

.77833 

47 

48 

.39540 

.65399 

.40939 

.69318 

.42357 

.73481 

.43792 

.77910 

48 

49 

.39563 

.65462 

.40963 

.69385 

.42381 

.73552 

.43816 

.77986 

49 

50 

.39586 

.65526 

.40986 

.69452 

.42404 

.73624 

.43840 

.78062 

50 

51 

.39610 

.65589 

.41010 

.69520 

.42428 

.73696 

.43864 

.78138 

51 

52 

.39633 

.65653 

.41033 

.69587 

.42452 

.73768 

.43888 

.78215 

62 

53 

.39656 

.65717 

.41057 

.69655 

.42476 

.73840 

.43912 

.78291 

53 

54 

.39679 

.65780 

.41080 

.69723 

.42499 

.73911   .43936 

.78368 

54 

55 

.39702 

.65844 

.41104 

.69790 

.42523 

.73983  I 

.43960 

.78445 

55 

56 

.39726 

.65908 

.41127 

.69858 

.42547 

74056  i  .43984 

.78521 

56 

57 

.39749 

.65972 

.41151 

.69926 

.42571 

.74128  I 

.44008 

.78598 

57 

58 

.39772 

.66036 

.41174 

.69994 

.42595 

.74200 

.44032 

.78675 

58 

59 

.39795 

.66100 

.41198 

.70062 

.42619 

.74272 

.44057 

.78752 

59 

60 

.39819 

.66164 

.41221 

.70130 

.42642 

.74345  ,1  .44081 

.78829 

60 

TABLE  XXII.— VEKS1NES  AND   EXSECANTS. 


373 


56° 

57° 

58° 

59° 

Vers. 

Exsec. 

Vers. 

Exsec.  |  Vers. 

Exsec. 

Vers. 

Exsec. 

o 

.44081 

.78829 

.45536 

.83608   .47008 

.88708 

.48496 

.94100 

0 

1 

.44105 

.78906 

.45500 

.83690   .47033 

.88796 

:  .48521 

.94254 

1 

2 

.44129 

.78984 

.4.5585 

.83773  1  .47057 

.88884 

.48546 

.94349 

2 

3 

.44153 

.79001 

.45009 

.83855   .4J082 

.88972 

.48571 

.94443 

3 

4 

.44177 

.79138 

.45634 

.83938 

.47107 

.89060 

.48596 

.94537 

4 

5 

.44201 

.79216 

.45058 

.84020 

.47131 

.89148 

.48621 

.94632 

5 

6 

.44225 

.79293 

.45083 

.84103  1  .47156 

.89237 

.48646 

.94726 

6 

7 

.44250 

.79371 

.45707 

.84186  j  .47181 

.89325 

.48671 

.94821 

7 

8 

.44274 

.79449 

.45731 

.84209   .47206 

.89414 

.48696 

.94916 

8 

9 

.44298 

.79527 

.45756 

.84352   .47230 

.89503 

.48721 

.95011 

9 

10 

.44322 

.79604 

.45780 

.84435 

.47255 

.89591 

.48746 

.95106 

10 

11 

.44346 

.79682 

.45805 

.84518 

.47280 

.89680 

.48771 

.95201 

11 

12 

.44370 

.79761 

.45829 

.84601 

.47304 

.89769 

.48796 

.95296 

12 

13 

.44395 

.79839 

.45854 

.84685 

.47329 

.89858 

.48821 

.95392 

13 

14 

.44419 

.79917 

.45878 

.84708 

.47354 

.89948 

.48846 

.95487 

14 

15 

.44443 

.79995 

.45903 

.84852 

.47379 

.90037 

.48871 

.95583 

15 

16 

.44407 

.80074 

.45927 

.84935 

.47403 

.90126 

.48896 

.95678 

16 

17 

.44491 

.80152 

.45951 

.85019 

.47428 

.90216 

.48921 

.95774 

17 

18 

.44516 

.80231 

.45976 

..85103 

.47453 

.90305 

.48946 

.95870 

18 

19 

.44540 

.80309 

.46000 

.85187 

.47478 

.90395 

.48971 

.95966 

19 

20 

.44504 

.80388 

.40025 

.85271 

.47502 

.90485 

.48998 

.96062 

20 

21 

.44588 

.80467 

.46049 

.85355 

.47527 

.90575 

.49021 

.96158 

21 

oo 

.44612 

.80546 

.46074 

.85439  j!  .47552 

.90605 

.49046 

.96255 

22 

23 

.44637 

.80625 

.46098 

.85523 

.47577 

.90755 

.49071 

.96351 

23 

24   .41601 

.80704 

.40123 

.85608 

.47601 

.90845 

.49096 

.96448 

24 

25 

.44685 

.80783 

.40147 

.85692 

.47626 

.90935 

.49121 

.96544 

25 

26 

.44709 

.80862 

.46172 

.85777 

.47051 

.91026 

.49146 

.96641 

26 

27 

.44734 

.80942 

.46196 

.85861 

.47676 

.91116 

.49171 

.96738 

27 

28 

.44758 

.81021 

.46221 

.85946 

.47701 

.91207 

.49196 

.96835 

28 

29 

.44782 

.81101 

.46246 

.86031 

.47725 

.91297 

.49221 

.96932 

29 

30 

.44806 

.81180 

.46270 

.86116 

.47750 

.91388 

.49246 

.97029 

30 

31 

.44831 

.81260 

.46295 

.86201 

.47775 

.91479 

.49271 

.97127 

31 

32 

.44855 

.81340 

.46319 

.86286   .47800 

.91570 

.49296 

.97224 

32 

33 

.44879 

.81419 

.40344 

.86371 

.47825 

.91061 

.49321 

.97322 

33 

34 

.44903 

.81499 

.46368 

.86457 

.47849 

.91758 

.49346 

.97420 

34 

35 

.44928 

.81579 

.46393 

.86542 

.47874 

.91844 

.49372 

.97517 

35 

36 

.44952 

.81659 

.46417 

.86627 

.47899 

.91935 

.49397 

.97615 

36 

37 

.44976 

.81740 

.46442 

.86713 

.47924 

.92027 

.49422 

.97713 

37 

38 

.45001 

.81820 

.46466 

.86799 

.47949 

.92118 

.49447 

.97811 

38 

39 

.45025 

.81900 

.46491 

.86885 

.47974 

.92210 

.49472 

.97910 

39 

40 

.45049 

.81981 

.46516 

.80990 

.47998 

.92302 

.49497 

.98008 

40 

41 

.45073 

.82061 

.46540 

.87056 

.48023 

.92394 

.49522 

.98107 

41 

42 

.45098 

.82142 

.46565 

.87142 

.48048 

.92486 

.49547 

.98205 

42 

43 

.45122 

.82222 

.46589 

.87229 

.48073 

.92578 

.4957'2 

.98304 

43 

44 

.45146 

.82303 

.46614 

.87315 

.48098 

.92670 

.49597 

.98403 

44 

45 

.45171 

.82384 

.46639 

.87401 

.48123 

.92762 

.49623 

.98502 

45 

46 

.45195 

.82465 

.46663 

.87488 

.48148 

.92855 

.49648 

.98601 

46 

47 

.45219 

.82546 

.46688 

.87574 

.48172 

.92947 

.49673 

.98700 

47 

48 

.45244 

.82627 

.46712 

.87661 

.48197 

.93040 

.49698 

.98799 

48 

49 

.45268 

.82709 

.46737 

.87748 

.48222 

.93133 

.49723 

.98899 

49 

50 

.45292 

.82790 

.46762 

.878:34 

.48247 

.93226 

.49748 

.98998 

50 

51 

.45317 

.82871 

.46786 

.87921 

.48272 

.93319 

.49773 

.99098 

51 

52 

.45341 

.82953 

.46811 

.88008 

.48237 

.93412 

.49799 

.99198 

52 

53 

.45305 

.83034 

.408:36 

.88095  1 

.48322 

.93505 

.49824 

.99298 

53 

54 

.45390 

.83116 

.46800 

.88183 

.48347 

.93598 

.49849 

.99398 

54 

55 

.45414 

.a3198 

.46885 

.88270 

.48372 

.93692 

.49874 

.99498 

55 

56 

.45439 

.83280 

.46909 

.88357 

.48396 

.93785 

.49899 

.99598 

56 

57 

.45403 

.83362 

.46934 

.88445 

.48421 

.93879 

.49924 

.99698 

57 

58 

.45487 

.83444 

.46959 

.88532 

.48446 

.93973 

.49950 

.99799 

58 

59 

.45512 

.83526 

.46983 

.88620 

.48471 

.94066 

.49975 

.99899 

59 

60   .45536 

.83608 

.47008 

.88708  i 

.48496 

.94160 

.50000   .00000 

60 

374 


TABLE  XXII.-VERS1XES  AND   EXSECANTS. 


/ 

i 

c 

0° 

6 

i« 

!  « 

2° 

e 

3° 

Vers. 

Exsec. 

Vers. 

Exsec. 

Yers. 

Exsec. 

Vers. 

Exsec. 

0 

.50000 

1.00000 

.51519 

1.06267 

.53053 

.13005 

.54GOl" 

.20269 

0 

1 

.50025 

1.00101 

.51544 

1.06375 

.53079 

.13122 

.54627 

.20395 

1 

8 

.50050 

1.00202 

.51570 

1.06483 

.53104 

.13239 

.54653 

.20521 

2 

I 

.50076 

1.00303 

.51595 

.06592 

.53130 

.13356 

.54679 

.20647 

3 

4 

.50101 

1.00404 

.51621 

.06701 

.53156 

.13473 

.54705 

.20773 

4 

5 

.50126 

1.00505 

.51646 

.06809 

.53181 

.13590 

.54731 

.20900 

B 

6 

.50151 

1.00607 

.51672 

.06918 

.53207 

.13707 

.54757 

.21026 

6 

7 

.50176 

1.00708 

.51697 

.07027 

.53233 

.13825 

.54782 

.21153 

7 

B 

.50202 

1.00810 

.51723 

.07137 

.53258 

.13942 

.54808 

.21280 

B 

'.» 

.50227 

1.00912 

.51748 

.07246 

.53284 

.14060 

.54834 

.21407 

0 

10 

.50252 

1.01014 

.51774 

.07356 

.53310 

.14178 

.54860 

.21535 

10 

11 

.50277 

1.01116 

.51799 

.07465 

.53336 

.14296 

.54886 

.21662 

11 

19 

.50303 

1.01218 

.51825 

.07575 

.5:3361 

.14414 

.54912 

.21790 

12 

18 

.50328 

1.01320 

.51850 

.07685 

.53387 

.14533 

.54938 

.21918 

13 

14 

.50353 

1.01422 

.51876 

.07795 

.53413 

.14651 

.54964 

.22045 

14 

15 

.50378 

1.01525 

.51901 

.07905 

.5.3439 

.14770 

.54900 

.22174 

15 

16 

.50404 

1.01628 

.51927 

.08015 

.53464 

.14889 

.55016 

.22302 

16 

17 

.50429 

1.01730 

.51952 

.08126 

.53490 

.15008 

.55042 

.22430 

17 

18 

.50454 

1.01833 

.51978 

.08236 

.53516 

.15127 

.55068 

.22559 

IS 

19 

.50479 

1.01936 

.52003 

.08347 

.53342 

.15246 

.55094 

.22688 

& 

90 

.50505 

1.02039 

.52029 

.08458 

.53567 

.15306 

.55120 

.22817 

20 

21 

.50530 

1.02143 

.52054 

.08569 

.53593 

.15485 

.55146 

.22946 

21 

32 

.50555 

1  02246 

.52080 

.08680 

.53619 

.15605 

.55172 

.23075 

22 

28 

.50581 

1.02349 

.52105 

.08791 

.53645 

.15725 

.55198 

.23205 

23 

•34 

.50606 

1.02453 

.52131 

.08903 

.53670 

.15845 

.55224 

.23334 

24 

26 

.50631 

1.02557 

.52156 

.09014 

.53696 

.15965 

.55250 

.23464 

25 

{6 

.50656 

1.02661 

.52182 

.09126 

.53722 

.16085 

.55276 

.23594 

26 

87 

.50682 

1.02765 

.52207 

.09238 

.53748 

.16206 

.55302 

.23724 

27 

28 

.50707 

1.02869 

.52233 

.09350 

.53774 

.16326 

.55328 

.23855 

28 

89 

.50732 

1.02973 

.52239 

.09462 

.53799 

.16447 

.55354 

.23985 

29 

to 

.50758 

1.03077 

.52284 

.09574 

.53825 

.16568 

.55380 

.24116 

SO 

31 

.50783 

1.03182 

.52310 

.09686 

.53851 

.16689 

.55406 

.24247 

31 

82 

.50808 

1.03286 

.52335 

.09799 

.53877 

.16810 

.55432 

.24378 

32 

88 

.50834 

1.03391 

.52361 

.09911 

.53903 

.16932 

.55458 

.24509 

3:5 

84 

.50859 

1.03496 

.52386 

.10024 

.53928 

.17053 

.55184 

.24640 

34 

88 

.50884 

1.03601 

.52412 

.10137 

.53954 

.17175 

.55510 

.24772 

35 

86 

.50910 

1.03706 

.52438 

.10250 

.53980 

.17297 

.55536 

.24903 

86 

37 

.50935 

1.03811 

.52463 

.10363 

.54006 

.17419 

.55563 

.25035 

37 

88 

.50960 

1.03916 

.52489 

.10477 

.54032 

.17541 

.55589 

25167 

38 

89 

.50986 

1.04022 

.52514 

.10590 

.54058 

.17663 

.55615 

.25300 

89 

40 

.51011 

1.04128 

.52540 

.10704 

.54083 

.17786 

.55641 

.25432 

40 

41 

.51036 

1.04233 

.52566 

.10817 

.54109 

.17909 

.55667 

.25565 

41 

42 

.51062 

1.04339 

.52591 

.10931 

.54135 

.18031 

.55693 

.25697 

42 

43 

.51087 

1.04445 

.52617 

.11045 

.54161 

.18154 

.55719 

.25830 

43 

44 

.51113 

1.04551 

.52642 

.11159 

.54187 

:  .18277 

.55745 

.25963 

44 

45 

.51138 

1.04658 

.52668 

.11274 

.54213 

.18401 

.55771 

.26097 

45 

46 

.51163 

1.04764 

.52694 

.11388 

.54238 

.18524 

.55797 

.26230 

46 

47 

.51189 

1.04870 

.52719 

.11503 

.54264 

.18648 

.55823 

.26364 

47 

48 

.51214 

1.04977 

.52745 

.11617 

.54290 

.18772 

.55849 

.26498 

4S 

4!) 

.51239 

1.05084 

.52771 

.11732 

.54316 

.18895 

.55876 

.26632 

4!) 

5U 

.51265 

1.05191 

.52796 

.11847 

.54342 

.19019 

.55902 

.26766 

50 

51 

.54290 

1.05298 

.52822 

.11963 

.54368 

.19144 

.55928 

.26900 

Bl 

52 

.51316 

1.05405 

.52848 

.12078 

.54394 

.19268 

.55954 

.27035 

52 

58 

.51341 

1.05512 

.52873 

.12193 

.54420 

.19393 

.55980 

.27169 

58 

54 

.51366 

1.05619 

.52899 

.12309 

.54446 

.19517 

.56006 

.27304 

51 

55 

.51392 

1.05727 

.52924 

.12425 

.54471 

.19642 

.56032 

.27439 

66 

56 

.51417 

1.05835 

.52950 

.12540 

.54497 

19767 

.56058 

.27574 

56 

57 

.51443 

1.05942 

.52976 

.12657  i 

.54523 

19892 

.5(5084 

.27710 

57 

58 

.51468 

1.06050 

.53001 

.12773 

.54549 

20018 

.56111 

.27845 

58 

59 

.51494 

1.06158 

.53027 

:  .12889 

.54575 

20143 

.5(5137 

.27981 

59 

60 

.51519 

1.06267 

.53053 

.13005 

.54601 

20269  i 

.56163 

.28117 

60 

TABLE   XXII. — VEKSINES  AND   EXSECANTS. 


6 

40 

6 

5° 

!     i 

6° 

6 

7° 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.56163 

.28117 

.57738 

1.36620 

.59326 

1.45859 

.60927 

1.55930 

1 

.56189 

.28253 

.57765 

.36768 

.59353 

1.46020 

.60954 

1.56106 

2 

.56215 

.28390 

.57791 

.36916 

.59379 

1.46181 

.60980 

1.56282 

3 

.56241 

.28526  ! 

.57817 

.37064 

.59406 

1.46342 

.61007 

1.56458 

4 

.56267 

.28663  i 

.57844 

.37212 

.59433 

1.46504 

.61034 

1.56684 

5 

.56294 

.28800 

.57870 

.37361 

.59459 

1.46665 

.61061 

1.56811 

G 

.56320 

.28937 

.57896 

.37509 

.59486 

1.46827 

.61088 

1.56988 

7 

.56346 

.29074 

.57923 

.37658  i 

.59512 

1.46989 

.61114 

1.57165 

8 

.56372 

.29211  ! 

.57949 

.37808  : 

.59539 

1.47152 

.61141 

1.57342 

g 

.56398 

.29349  i 

.57976 

.37957  : 

.59566 

1.47314 

.61168 

1.57520 

10 

.56425 

1.29487  i 

.58002 

.3810? 

.59592 

1.47477 

.61195 

1.57698 

11 

.56451 

1.29625 

.58028 

.38256 

.59619 

1.47640 

.61222 

1.57876 

12 

.56477 

1-29763 

.58055 

.38406 

.59645 

1.47804 

.61248 

1.58054 

13 

.56503 

1.29901 

.58081 

.38556 

.59672 

1.47967 

.61275 

1.58233 

14 

.56529 

1.30040 

.58108 

.38707  i 

.59699 

1.48131 

.61302 

1.58412 

15 

.56555 

1.30179 

.58134 

.38857 

.59725 

1.48295 

.61329 

1.58591 

16 

.56582 

1.30318 

.58160 

.39008 

.59752 

1.48459 

.61356 

1.58771 

17 

.56608 

1.30457 

.58187 

.39159 

.59779 

1.48624 

.61383 

1.58950 

IS 

.56634 

.30596 

.58213 

.39311 

.59805 

1.48789 

.61409 

1.59130 

19 

.56660 

.30735 

.58240 

.39462 

.59832 

1.48954 

.61436 

1.59311 

20 

.56687 

.30875 

.58266 

.39614 

.59859 

1.49119  i 

.61463 

1.59491 

21 

,56713 

.31015 

.58293 

.39766 

.59885 

1.49284 

.61490 

1.59672 

22 

.56739 

.31155 

.58319 

.39918 

.59912 

1.49450 

.61517 

1.59853 

23 

.56765 

1.31295 

.58345 

.40070 

.59938 

1.49616 

.61544 

1.60035 

24 

.56791 

1.31436 

.58372 

.40222  | 

.59965 

1.49782 

.61570 

1.60217 

25 

.56818 

1.31576 

.58398 

.4037'5  | 

.59992 

1.49948 

.61597 

1.60399 

26 

.56844 

1.31717 

.58425 

.40528 

.60018 

1.50115 

.61624 

1.60581 

27 

.56870 

1.31858 

.58451 

.40681 

.60045 

1.50282 

.61651 

1.60763 

38 

.56896 

1.31999 

.58478 

.40835 

.60072 

1.50449  i 

.61678 

1.60946 

29 

.56923 

1.32140 

.58504 

.40988 

.60098 

1.50617 

.61705 

1.61129 

30 

.56949 

1.32282 

.58531 

.41142 

.60125 

1.50784 

.61732 

1.61313 

31 

.56975 

1.32424 

.58557 

.41296 

.60152 

1.50952 

.61759 

1.61496 

32 

.57001 

1.32566 

.58584 

.41450 

.60178 

1.51120 

.61785 

1.61680 

33 

.57028 

1.32708 

.58610 

.41605 

.60205 

1.51289  , 

.61812 

1.61864 

34 

.57054 

1.32850 

.58637 

.41760 

.60232 

1.51457 

.61839 

1.62049 

35 

.57080 

1.32993 

.58663 

.41914 

.60259 

1.51626 

.61866 

1.62234 

30 

.57106 

1.33135 

.58690 

.42070 

.60285 

1.51795 

.61893 

1.62419 

37 

.57133 

1.33278 

.58716 

.42225 

.60312 

1.51965  : 

.61920 

1.62604 

38 

.57159 

1..  33422 

.58743 

.42380 

.60339 

1.52134 

.61947 

1.62790 

39 

.57185 

1.33565 

.58769 

.42536 

.60365 

1.52304 

.61974 

1.62976 

40 

.57212 

1.33708 

.58796 

.42692 

.60392 

1.52474 

.62001 

1.63162 

41 

.57238 

1.33852 

.58822 

.42848 

.60419 

1.52645 

.62027 

1.63348 

42 

.57264 

1.33996 

.58849 

.43005 

.60445 

1.52815  j 

.62054 

1.63535 

43 

.57291 

1.34140 

.58875 

.43162 

.60472 

1.52986 

.62081 

1.63722 

44 

.57317 

1.34284 

.58902 

.43318 

.60499 

1.53157 

.62108 

1.03909 

45 

.57343 

1.34429 

.58928 

.43476 

.60526 

1.53329 

.62135 

1.64097 

46 

.57369 

1.34573 

.58955 

.43633 

.60552 

1.53500 

.62162 

1.64285 

47 

.57396 

1.34718 

.58981 

.43790 

.60579 

1.53672 

.62189 

1.64473 

48 

.57422 

1.34863 

.59008 

.43948 

.60606 

1.53845  ] 

.62216 

1.64662 

49 

.57448 

1.35009 

.59034 

.44106 

.60633 

1.54017 

.62243 

1.64851 

50 

.57475 

1.35154 

.59061 

.44264 

.60659 

1.54190 

.62270 

1.65040 

51 

.57501 

1.35300 

.59087 

.44423 

.60686 

1.54363 

.62297 

1.65229 

52 

.57527 

1.35446 

.59114 

.44582 

.60713 

1.54536 

.62324 

1.65419 

58 

.57554 

1.35592 

.59140 

.44741 

.60740 

1.54709 

.62351 

1.65609 

54 

.57580 

1.35738 

.59167 

.44900 

.60766 

1.54883 

.62378 

1.65799 

55 

.57606 

1.35885 

.59194 

.45059 

.60793 

1.5505? 

.62405 

1.65989 

56 

.57633 

1.36031 

.59220 

.45219 

.60820 

1.55231 

.62431 

1.66180 

57 

.57659 

J.  36173 

.59247 

1.45378 

.60847 

1.55405 

.62458 

1.66371 

58 

.57685 

1.36325 

.59273 

1.45539 

.60873 

1.55580 

.62485 

1.66563 

59 

.57712 

1.36473 

.59300 

1.45099 

.60900 

1.55755 

.62512 

1.66755 

60 

.57738 

1.36620 

.59326 

1.45859 

.60927 

1.55930 

.62539 

1.66947 

376 


TABLE   XXII.-VERSINES  AND   EXSECANTS. 


/ 

38' 

( 

59° 

\ 

ro° 

•i 

1° 

Vers, 

Exsec. 

Vers. 

Exsec. 

1  Vers. 

Exsec. 

Vers. 

Exsec. 

1 

.62539 

1.66947 

.64163 

1.79043 

.65798 

1.92380 

.67443 

2.07155 

0 

• 

.62566 

1.67139 

.64190 

1.79254 

i  .65825 

1.92614 

.67471 

2.07415 

< 

.62593 

1.67332 

.64218 

1.79466 

.65853 

1.92849 

.67498 

2.07675 

2 

] 

.62620 

1.67525 

.64245 

1.79679 

.65880 

1.93083 

.67526 

2.07936 

g 

/ 

.62647 

1.67718 

.64272 

1.79891 

.65907 

1.93318 

.67553 

2.08197 

i 

5 

.62674 

1.67911 

.64299 

1.80104 

.65935 

1.93554 

.67581 

2.08459 

i 

6 

.62701 

1.68105 

.64326 

1.80318 

.65962 

1.93790 

.67608 

2.08721 

( 

5 

.62728 

1.68299 

.64353 

1.80531 

1  .65989 

1.94026 

.67636 

2.08983 

7 

8 

.62755 

1.68494 

.64381 

1.80746 

.66017 

1.94263 

.67663 

2.09246 

8 

9 

.62782 

1.68689 

.64408 

1.80960 

:  .66044 

1.94500 

.67691 

2.09510 

9 

10 

.62809 

1.68884 

.64435 

1.81175 

.66071 

1.94737 

.67718 

2.09774 

10 

11 

.62836 

1  69079 

.64462 

1.81390 

.66099 

1.94975 

.67746 

2.10038 

11 

12 

.62863 

1.69275 

.64489 

1.81605 

.66126 

1.95213 

.67773 

2.10303 

12 

13 

.62890 

1.69471 

.64517 

1.81821 

.66154 

1.95452 

.67801 

2.10568 

13 

14 

.62917 

1.69667 

.64544 

1.82037 

.66181 

1.95691 

.67829 

2.10834 

14 

15 

.62944 

1.69864 

.64571 

1.82254 

.66208 

1.95931 

.67856 

2.11101 

15 

16 

.62971 

1.70061 

.64598 

1.82471 

.66236 

1.96171 

.67884 

2.11367 

16 

17 

.62998 

1.70258 

.64625 

1.82688 

.66263 

1.96411 

.67911 

2.11635 

17 

18 

.63025 

1.70455 

.64653 

1.82906 

.66290 

1.96652 

.67939 

2.11903 

18 

19 

.63052 

1.70653 

.64680 

1.83124 

.66318 

1.96893 

.G7966 

2.12171 

19 

20 

.63079 

1.70851 

.64707 

1.83342 

.66345 

1.97135 

.67994 

2.12440 

20 

21 

.63106 

1.71050 

.64734 

1.83561 

.66373 

1.97377 

.68021 

2.12709 

21 

22 

.63133 

1.71249 

.64761 

1.83780 

.66400 

1.97619 

'  .68049 

2.12979 

22 

23 

.63161 

1.71448 

.64789 

1.83999 

.66427 

1.97862 

.68077 

2.13249 

23 

24 

.63188 

1.71647 

.64816 

1.84219 

.66455 

1.98106 

.68104 

2.13520 

24 

25 

.63215 

1.71847 

.64843 

1.84439 

!  .66482 

1.98349 

.68132 

2.13791 

25 

26 

.63242 

1.72047 

.64870 

1.84659 

I  .66510 

1.98594 

.68159 

2.14063 

26 

27 

.63269 

1.72247 

.64898 

1.84880 

.66537 

1.98838 

.68187 

2.14335 

27 

28 

.63296 

1.72448 

.64925 

1.85102 

.66564 

1.99083 

.68214 

2.14608 

28 

29 

.63323 

1.72649 

.64952 

1.85323 

.66592 

1.99329 

.68242 

2.14881 

29 

30 

.63350 

1.72850 

.64979 

1.85545 

.66619 

1.99574 

.68270 

2.15155 

30 

31 

.63377 

1.73052 

.65007 

1.85767 

.66647 

1.99821 

.68297 

2.15429 

31 

32 

.63404 

1.73254 

.65034 

1.85990 

.66674 

2.00067 

.68325 

2.15704 

32 

33 

.63431 

1.73456 

.65061 

1.86213 

.66702 

2.00315 

.68352 

2.15979 

33 

34 

.63458 

1.73659  1 

.65088 

1.86437 

.66729 

2.00562 

.68380 

2.16255 

34 

35 

.63485 

1.73862 

.65116 

1.86661 

.66756 

2.00810 

.68408 

2.16531 

35 

36 

.63512 

1.74065 

.65143 

1.86885 

.66784 

2.01059 

.68435 

2.16808 

36 

37 

.63539 

1.74269 

.65170 

1.87109 

.66811 

2.01308 

.68463 

2.17085 

37 

38 

.63566 

1.74473 

.65197 

1.87334 

j  .66839 

2.01557 

.68490 

2.17363 

38 

39 

.63594 

1.74677 

.65225 

1.87560 

.66866 

2.01807 

.68518 

2.17641 

39 

40 

.63621 

1.74881 

.65252 

1.87785 

.66894 

2.02057 

.68546 

2.17920 

40 

41 

.63648 

1.75086 

.65279 

1.88011 

.66921 

2.02308 

.68573 

2.18199 

41 

42 

.63675 

1.75292 

.65306 

1.88238 

.66949 

2.02559 

.68601 

2.18479 

42 

43 

.63702 

1.75497 

.65334 

1.88465 

.6697'6 

2.02810 

.68628 

2.18759 

43 

44 

.63729 

1.75703 

.65361 

1.88692 

.67003 

2.03062 

.68656 

2.19040 

44 

45 

.63756 

1.75909 

.65388 

1.88920 

.67031 

2.03315 

.68684 

2.19322 

45 

46 

.63783 

1.76116 

.65416 

1.89148 

.67058 

2.03568 

.68711 

2.19604 

46 

47 

.63810 

1.76323 

.65443 

1.89376 

.67086 

2.03821 

.68739 

2.19886 

47 

48 

.63838 

1.76530  j 

.65470 

1.89605 

.67113 

2.04075 

.68767 

2.20169 

48 

49 

.63865 

1.76737 

.65497 

1.89834 

.67141 

2.04329 

.68794 

2.20453 

49 

50 

.63892 

1.76945 

.65525 

1.90063 

.67168 

2.04584 

.68822 

2.20737 

50 

51 

.63919 

1.77154 

.65552 

1.90293 

.67196 

2.04839 

.68849 

2.21021 

51 

52 

.63946 

1.77362 

.65579 

1.90524 

.67223 

2.05094 

.68877 

2.21306 

52 

53 

.63973 

1.77571  ' 

.65607 

1.90754 

.67251 

2.05350 

.68905 

2.21592 

53 

54 

.64000 

1.77780  , 

.65634 

1.90986 

.67278 

2.05607 

.68932 

2.21878 

54 

55 

.64027 

1.77990 

.65661 

1.91217 

.67306 

2.05864 

.68960 

2.22165 

55 

56 

.64055 

1.78200 

.65689 

1.91449 

.67333 

2.06121 

.68988  | 

2.22452 

56 

57 

64082 

1.78410 

.65716 

1.91681 

.67361 

2.06379 

.69015 

2.22740 

57 

58 

.64109 

1.78621 

.65743 

1.91914 

.67388 

2.0663? 

.69043 

2.23028 

58 

59 

.64136 

1.78832 

.65771 

1.92147 

.67413 

Xi.lKJWMi 

.(59071 

2.23317 

59 

60 

.64163 

1.79043 

.65798 

1.92380  I 

.67443 

2.07155 

.69098 

2.23607 

60 

TABLE  XXII.— VERSINES  AND  EXSECANTS. 


' 

72° 

73° 

740 

75° 

,/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.69098 

2.23607 

.70763 

2.42030 

.72436 

2.62796 

.74118 

2.86370 

0 

1 

.69126 

2.23897 

.70791 

2.42356 

.72464 

2.63164 

.74146 

2.86790 

1 

2 

.69154 

2.24187 

.70818 

2.42683 

.72492 

2.63533 

.74174 

2.87211 

2 

3 

.69181 

2.24478 

.70846 

2.43010 

.72520 

2.63903 

.74202 

2.87633 

3 

4 

.69209 

2.24770 

.70874 

2.43337 

.72548 

2.64274 

.74231 

2.88056 

4 

5 

.69237 

2.25062 

.70902 

2.436G6 

.72576 

2.64645 

.74259 

2.88479 

5 

C 

69264 

2.25355 

.70930 

2.43995 

.72604 

2.65018 

.74287 

2.88904 

6 

7 

.69292 

2.25648 

.70958 

2.44324 

.72632 

2.65391 

.74315 

2.89330 

7 

8 

.69320 

2.25942 

.70985 

2.44655 

.72660 

2.65765 

.74343 

2.89756 

8 

B 

.69347 

2.26237 

.71013 

2.44986 

.72688 

2.66140 

.74371 

2.90184 

9 

10 

.69375 

2.26531 

.71041 

2.45317 

.72716 

2.66515 

.74399 

2.90613 

10 

11 

.69403 

2.26827 

.71069 

2.45650 

.72744 

2.66892 

.74427 

2.91042 

11 

12 

.69430 

2.27123 

.71097 

2.45983 

.72772 

2.67269 

.74455 

2.91473 

12 

18 

.69458 

2.27420 

.71125 

2.46316 

.72800 

2.67647 

.74484 

2.91904 

13 

14 

.69486 

2.27717 

.71153 

2.46651 

.72828 

2.68025 

.74512 

2.92337 

14 

15 

.69514 

2.28015 

.71180 

2.46986 

.72856 

2.68405 

.74540 

2.92770 

15 

16 

.69541 

2.28313 

.71208 

2.47321 

.72884 

2.68785 

.74568 

2.93204 

16 

17 

.69569 

2.28612 

.71236 

2.47658 

.72912 

2.69167 

.74596 

2.93640 

17 

18 

.69597 

2.28912 

.71264 

2.47995 

.72940 

2.69549 

.74624 

2.94076 

IS 

19 

.69624 

2.29212 

.71292 

2.48333 

.72968 

2.69931 

.74652 

2.94514 

19 

20 

.69652 

2.29512 

.71320 

2.48671 

.72996 

2.70315 

.74680 

2.94952 

20 

21 

.69680 

2.29814 

.71348 

2.49010 

.73024 

2.70700 

.74709 

2.95392 

21 

22 

.69708 

2.30115 

.71375 

2.49350 

.73052 

2.71085 

.74737 

2.95832 

•-.';! 

23 

.69735 

2.30418 

.71403 

2.49691 

.73080 

2.71471 

.74765 

2.96274 

23 

24 

.69763 

2.30721 

.71431 

2.50032 

.73108 

2.71858 

.74793 

2.96716 

24 

25 

.69791 

2.31024 

.71459 

2.50374 

.73136 

2.72246 

.74821 

2.97160 

25 

26 

.69818 

2.31328 

.71487 

2.50716 

.73164 

2.72635 

.74849 

2.97604 

26 

27 

.69846 

2.31633 

.71515 

2.51060 

.73192 

2.73024 

.74878 

2.98050 

27 

28 

.69874 

2.31939 

.71543 

2.51404 

.73220 

2.73414 

.74906 

2.98497 

88 

29 

.69902 

2,32244 

.71571 

2.51748 

.73248 

2.73806 

.74934 

2.98944 

•J'.i 

30 

.69929 

2.32551 

.71598 

2.52094 

.73276 

2.74198 

.74962 

2.99393 

80 

31 

.69957 

2.32858 

.71626 

2.52440 

.73304 

2.74591 

.74990 

2.99843 

SI 

32 

.69985 

2.33166 

.71654 

2.52787 

.73332 

2.74984 

.75018 

3.00293 

:w 

33 

.70013 

2.33474 

.71682 

2.53134 

.73360 

2.75379 

.75047 

3.00745 

88 

34 

.70040 

2.33783 

.71710 

2.53482 

.73388 

2.75775 

.75075 

3.01198 

34 

35 

.70068 

2.34092 

.71738 

2.53831 

.73416 

2.76171 

.75103 

3.01652 

35 

36 

.70096 

2.34403 

.71766 

2.54181 

.73444 

2.76568 

.75131 

3.02107 

86 

37 

.70124 

2.34713 

.71794 

2.54531 

.73472 

2.76966 

.75159 

3.02563 

87 

38 

.70151 

2.35025 

.71822 

2.54883 

.73500 

2.77365 

.75187 

3.03020 

88 

39 

.70179 

2.35336 

.71850 

2.55235 

.73529 

2.77765 

.75216 

3.03479 

:«) 

40 

.70207 

2.35649 

.71877 

2.55587 

.73557 

2.78166 

.75244 

3.03938 

40 

41 

.70235 

2.35962 

.71905 

2.55940 

.73585 

2.78568 

.75272 

3.04398 

41 

42 

.70263 

2.36276 

.71933 

2.56294 

.73613 

2.78970 

.75300 

3.04860 

42 

43 

.70290 

2.36590 

.71961 

2.56649 

.73641 

2.79374 

.75328 

3.05322 

43 

44 

.70318 

2.36905 

.71989 

2.57005 

.73669 

2.79778 

.75356 

3.05786 

44 

45 

.70346 

2.37221 

.72017 

2.57361 

.73697 

2.80183 

.75385 

3.06251 

45 

46 

.70374 

2.37537 

.72045 

2.57718 

.73725 

2.80589 

.75413 

3.06717 

46 

47 

.70401 

2.37854 

.72073 

2.58076 

.73753 

2.80996 

.75441 

3.07184 

47 

48 

.70429 

2.38171 

.72101 

2.58434 

.73781 

2.81404 

.75469 

3.07652 

48 

49 

.70457 

2.38489 

.72129 

2.58794 

.73809 

2.81813 

.75497 

3.08121 

49 

50 

.70485 

2.38808 

.72157 

2.59154 

.73837 

2.82223 

.75526 

3.08591 

50 

51 

.70513 

2.39128 

.72185 

2.59514 

.73865 

2.82633 

.75554 

3.09063 

51 

52 

.70540 

2.39448 

.72213 

2.59876 

.73893 

2.83045 

.75582 

3.09535 

52 

53 

.70568 

2.39768 

.72241 

2.60238 

.73921 

2.83457 

.75610 

3.10009 

58 

54 

.70596 

2.40089 

.72269 

2.60601 

.73950 

2.83871 

.75639 

3.10484 

54 

55 

.70624 

2.40411 

.72296 

2.60965 

.73978 

2.84285 

.75667 

3.10960 

55 

56 

.70652 

2.40734 

.72324 

2.61330 

.74006 

2.84700 

.75695 

3.11437 

5G 

57 

.70679 

2.41057 

.72352 

2.61695 

.74034 

2.85116 

.75723 

3.11915 

57 

58 

.70707 

2.41381 

.72380  2.62061 

.74062 

2.855J33 

.75751 

3.12394 

58 

59 

.70735 

2.41705 

.72408 

2.62428 

.74090 

2.85951 

.75780 

3.12875 

59 

60 

.70763 

2.42030 

.72436 

2.62796 

.74118 

2.86370 

.75808 

3.13357 

60 

TABLE  XXtL-VERSlNES  AND  EXSEC  A  NTS. 


' 

76° 

77° 

78° 

79° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec, 

Vers. 

Exsec. 

Vers; 

Exsec, 

0 

.75808 

3.13357 

.77505 

3.44541 

.79209 

3.80973 

.80919 

4.24084 

0 

1 

.75836 

3.13839 

.77533 

3.45102 

.79237, 

3.81633 

.80948 

4.24870 

1 

2 

.75864 

3.14323 

.77562 

3.45664 

.79266 

3.82294 

.80976 

4.25658 

2 

3 

.75892 

3.14809 

.77590 

3.46228 

.79294 

3.82956 

.81005 

4.26448 

3 

4 

.75921 

3.15295 

.77618 

3.46793 

.79323 

8.83621 

.81033 

4.27241 

4 

5 

.75949 

3.15782 

.77647 

3.47360 

i  79351 

3.84288 

,81062 

4.28036 

5 

6 

.75977 

3.16271 

.77675 

3.47928 

;  79380 

3.84956 

,81090 

4.28833 

6 

7 

.76005 

3.16761 

.77703 

3.48498 

,79408 

3,85627 

.81119 

4.29634 

7 

8 

.76034 

3.17252 

.77732 

3.49069 

.79437 

3.86299 

.81148 

4.30436 

8 

9 

.76062 

3.17744 

.77760 

3.49642 

.79465 

3.86973 

.81176 

4.81241 

9 

10 

.76090 

3.18238 

.77788 

3.50216 

.79493 

3.87649 

.81205 

4.32049 

10 

11 

.76118 

3.18733 

.77817 

3.50791 

.79522 

3.88327 

.81233 

4.32859 

11 

12 

.76147 

3.19228 

.77845 

3.51368 

.79550 

3.89007 

.81262 

4.33671 

12 

13 

.76175 

3.19725 

.77874 

3.51947 

.79579 

3.89689 

.81290 

4.34486 

13 

14 

.76203 

3.20224 

.77902 

3.52527 

.79607 

3.90373 

.81319 

4.35304 

14 

15 

.76231 

3.20723 

.77930 

3.53109 

.79636 

3.91058 

.81348 

4.36124 

15 

16 

.76260 

3.21224 

.77959 

3.53692 

.79664 

3.91746 

.81376 

4.36947 

16 

1? 

.76288 

3.21726 

.77987 

3.54277 

.79693 

3.92436 

.81405 

4.37772 

17 

18 

.76316 

3.22229 

.78015 

3.54863 

.79721 

3.93128 

.81433 

4.38600 

18 

19 

.76344 

3.22734 

.78044 

3.55451 

.79750 

3.93821 

.81462 

4.39430 

19 

20 

.78373 

3.23239 

.78072 

3.56041 

.79778 

3.94517 

.81491 

4.40263 

20 

81 

.76401 

3.23746 

.78101 

3.56632 

.79807 

3.95215 

.81519 

4.41099 

21 

22 

.76429 

3.24255 

.78129 

3.57224 

.79835 

3.95914 

.81548 

4.41937 

22 

23 

.76458 

3.24764 

.78157 

3.57819 

.79864 

3.96616 

.81576 

4.42778 

23 

24 

.76486 

3.25275 

.78186 

3.58414 

.79892 

3.97320 

.81605 

4.43622 

24 

85 

.76514 

3.25787 

.78214 

3.59012 

.79921 

3.98025 

.81633 

4.44468 

25 

26 

.76542 

3.26300 

.78242 

3.59611 

.79949 

3.98733 

.81662 

4.45317 

26 

27 

.76571 

3.26814 

.78271 

3.60211 

.79978 

3.99443 

.81691 

4.46169 

27 

38 

.76599 

3.27330 

.78299 

3.60813 

.80006 

4.00155 

.81719 

4.47023 

28 

29 

.76627 

3.27847 

.78328 

3.61417 

.80035 

4.00869 

.81748 

4.47881 

29 

30 

.76655 

3.28366 

.78356 

3.62023 

.80063 

4.01585 

.81776 

4.48740 

SO 

31 

.76684 

3.28885 

.78384 

3.62630 

.80092 

4.02303 

.81805 

4.49603 

31 

3:2 

.76712 

3.29406 

.78413 

3.63238 

.80120 

4.03024 

.81834 

4.50466 

32 

33 

.76740 

3.29929 

.78441 

3.63849 

.80149 

4.03746 

.81862 

4.51337 

33 

34 

.76769 

3.30452 

.78470 

3.64461 

.80177 

4.04471 

.81891 

4.52208 

34 

35 

.76797 

3.30977 

.78498 

3.65074 

.80206 

4.05197 

.81919 

4.53081 

35 

36 

.76825 

3.31503 

.78526 

3.65690 

.80234 

4.05926 

.81948 

4.53958 

36 

87 

.76854 

3.32031 

.7'8555 

3.66307 

.80263 

4.06657 

.81977 

4.54837 

37 

38 

.76882 

3.32560 

.78583 

3.66925 

.80291 

4.07390 

.82005 

4.55720 

38 

39 

.76910 

3.33090 

.78612 

3.67545 

.80320 

4.08125 

.82034 

4.56605 

39 

40 

.78938 

3.33622 

.78640 

3.68167 

.80348 

4.08863 

.82063 

4.57493 

40 

41 

.76967 

3.34154 

.78669 

3.68791 

.80377 

4.09602 

.82091 

4.58383 

41 

42 

.76995 

3.34689 

.78697 

3.69417 

.80405 

4.10344 

.82120 

4.59277  142 

43 

.77023 

3.35224 

.78725 

3.70044 

.80434 

4.11088 

.82148 

4.60174  43 

44 

.77052 

3.35761 

.78754 

3.70673 

.80462 

4.11885 

.82177 

4.61073  1  44 

45 

.77080 

3.36299 

.78782 

3.71303 

.80491 

4.12583 

.82206 

4.61976 

45 

46 

.77108 

3.36839 

.78811 

3.71935 

.80520 

4.13334 

.82234 

4.62881 

46 

47 

.77137 

3.37380 

.78839 

3.72569 

.80548 

4.14087 

.82263 

4.63790 

47 

48 

.77165 

3.37923 

.78868 

3.73205 

.80577 

4.14842 

.82292 

4.64701  48 

49 

.77193 

3.38466 

.78896 

3.73843 

.80605 

4.15599 

.82320 

4.65616  49 

50 

.77222 

3.39012 

.78924 

3.74482 

.80634 

4.16359 

.82349 

4.66533  50 

51 

.77250 

3.39558 

.78953 

3.75123 

.80662 

4.17121 

.82377 

4.67454  51 

:,2 

.77278 

3.40106 

.78981 

3.75766 

.80691 

4.17886 

.82406 

4.68377  52 

53 

.77307 

3.40656 

.79010 

3.76411 

.80719 

4.18652 

.82435 

4.69304 

53 

.54 

.77335 

3.41206 

.79038 

3.77057 

.80748 

4.19421 

.82463 

4.70234 

54 

55 

.77363 

3.4175P 

.79067 

3.77705 

.80776 

4.20193 

.82492 

4.71166 

55 

66 

.77392 

S.  42315* 

.79095 

3.78355 

.80805 

4.20966 

.82521 

4.72102 

56 

57 

.77420 

S.  42867 

.79123 

3.79007 

.80&33 

4.21742 

.82549 

4.73041 

57 

58 

.77448 

3.43424 

.79152 

3.79661 

.80862 

4.22521 

.82578 

4.73983 

58 

59 

.77477 

3.43982 

.79180 

3.80316 

.80891 

4.23301 

.82607 

4.74929 

59 

60 

.77505 

3.44541 

.79209 

3.80973 

.80919 

4.24084 

.82635 

4.75877 

60 

TABLE  XXII.— VERSINES  AND  EXSECANTS. 


' 

80° 

81°          82° 

83° 

' 

Vers. 

Exsec. 

Vers. 

Exsec.  I  Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.82635 

4.75877 

.84357 

5.39245  1  .86083 

6.18530 

.87813 

7.20551 

0 

1 

.82664 

4.76829 

.84385 

5.40422 

.86112 

6.20020 

.87842 

7.22500 

1 

8 

.82692 

4.777'84 

.84414 

5.41602 

.86140 

6.21517 

.87871 

7.24457 

2 

3 

.82721 

4.78742 

.84443 

5.42787 

.86169 

6.23019 

.87900 

7.26425 

3 

4 

.82750 

4.79703 

.84471 

5.43977 

!  .86198 

6.24529 

.87929 

7.28402 

4 

5 

.82778 

4.80667 

.84500 

5.45171 

.86227 

6.26044 

.87957 

7.36388 

5 

6 

.82807 

4.81635 

.84529 

5.46369 

.86256 

6.27566 

.87986 

7.32384 

6 

7 

.82836 

4.82606 

.84558 

5.47572 

.86284 

6.29095 

.88015 

7.34390 

7 

8 

.82864 

4.83581 

.84586 

5.48779 

.86313 

6.30630 

.88044 

7.36405 

8 

9 

.82893 

4.84558 

.84615 

5.49991 

.86342 

6.32171 

.88073 

7.38431 

9 

10 

.82922 

4.85539 

.84644 

5.51208 

.86371 

6.33719 

.88102 

7.40466 

10 

11 

.82950 

4.86524 

.84673 

5.52429 

.86400 

6.35274 

.88131 

7.42511 

11 

12 

.82979 

4.87511 

.84701 

5.53655 

.86428 

6.36835 

.88160 

7.44566 

12 

13 

.83003 

4.88502 

.84730 

5.54886 

.86457 

6.38403 

.88188 

7.46632 

13 

14 

.83036 

4.89497 

.84759 

5.56121 

.86486 

6.39978 

.88217 

7.48707  14 

15 

.83065 

4.90495 

.84788 

5.57361 

.86515 

6.41560 

.88246 

7.50793  15 

16 

.83094 

4.91496 

.84816 

5.58606 

.86544 

6.43148 

.88275 

7.52889  '16 

ir 

.83122  4.92501 

.84845 

5.59855 

.86573 

6.44743 

.88304 

7.54996  17 

18 

.83151 

4.93509 

.84874 

5.61110 

.86601 

6.46346 

.88333 

7.57113  |18 

19 

.83180 

4.94521 

.84903 

5.62369 

.86630 

6.47955 

.88362 

7.59241  !19 

20 

.83208 

4.95536 

.84931 

5.63633 

.86659 

6.49571 

.88391 

7.61379 

20 

21 

.83237 

4.96555 

.84960 

5.64902 

.86688 

6.51194 

.88420 

7.63528 

21 

38 

.83266 

4.97577 

.84989 

5.66176 

.86717 

6.52825 

.88448 

7.65688 

22 

88 

.83294 

4.98603 

.85018 

5.67454 

.86746 

6.54462 

.88477 

7.67859 

23 

24 

.83323 

4.99633 

.85046 

5.68738 

.86774 

6.56107 

.88506 

7.70041 

24 

86 

.83352 

5.00666 

.85075 

5.70027 

.86803 

6  57759 

.88535 

7.72234 

25 

2Q 

.83380 

5.01703 

.85104 

5.71321 

.86832 

6.59418 

.88564 

7.74438 

26 

27 

.83409 

5.02743 

.85133 

5.7'2620 

.86861 

6.61085 

.88593 

7.76653 

27 

2* 

.83438 

5.03787 

.85162 

5.73924 

.86890 

6.62759 

.88622 

7.78880 

28 

99 

.83467 

5.04834 

.85190 

5.75233 

.86919 

6.64441 

.88651 

7.81118 

29 

30 

.83495 

5.05886 

.85219 

5.76547 

.86947 

6.66130 

.88680 

7.83367 

30 

31 

.83524 

5.06941 

.85248 

5.77866 

.86976 

6.67826 

.88709 

7.85628 

31 

82 

.83553 

5.08000 

.85277 

•5.79191 

.87005 

6.69530 

.88737 

7.87901 

32 

33 

.83581 

5.09062 

.85305 

5.80521 

.87034 

6.71242 

.88766 

7.90186 

33 

84 

.83610 

5.10129 

!85334 

5.81856 

.87063 

6.72962 

.88795 

7.92482 

34 

33 

.83639 

5.11199 

.85363 

5.83196 

.87092 

6.74689 

.88824 

7.94791 

35 

36 

.83667 

5.12273 

.85392 

5.84542 

.87120 

6.76424 

.88853 

7.97111 

36 

37 

.83696 

5.13350 

.85420 

5.85893 

.87149 

6.78167 

.88882 

7.99444 

37 

38 

.83725 

5.14432 

.85449 

5.87250 

.87178 

6.79918 

.88911 

8.01788 

38 

39 

.83754 

5.15517 

.85478 

5.88612 

.87207 

6.81677 

.88940 

8.04146 

39 

40 

.83782 

5.16607 

.85507 

5.89979 

.87236 

6.83443 

.88969 

8.06515 

40 

41 

.83811 

5.17700 

.85536 

5.91352 

.87265 

6.85218 

.88998 

8.08897 

41 

42 

.83840 

5.18797 

.85564 

5.92731 

.87294 

6.87001 

.89027 

8.11292 

42 

43 

.83868 

5.19898 

.85593 

5.94115 

.87322 

6.88792 

.89055 

8.13699 

43 

44 

.83897 

5.21004 

.85622 

5.95505 

.87351 

6.90592 

.89084 

8.16120 

44 

45 

.83926 

5.22113 

.85651 

5.96900 

.87380 

6.92400 

.89113 

8.18553 

45 

46 

.83954 

5.23226 

.85680 

5.98301 

.87409 

6.94216 

.89142 

8.20999 

46 

47 

.83983 

5.24343 

.85708 

5.99708 

.87438 

6.96040 

.89171 

8.23459 

47 

48 

.84012 

5.25464 

.85737 

6.01120 

.87467 

6.97873 

.89200 

8.25931  48 

49 

.84041 

5.26590 

.85766 

6.02538 

.87496 

6.99714 

.89229 

8.28417  49 

50 

.84069 

5.27719 

.85795 

6.03962 

.87524 

7.01565 

.89258 

8.30917  50 

51 

.84098 

5.28853 

.85823 

6.05392 

.87553 

7.03423 

.89287  ' 

8.33430  fel 

fig 

.84127 

5.29991 

.85852 

6.06828 

.87582 

7.05291 

.89316 

8.35957  :52 

53 

.84155 

5.31133 

.85881 

6.08269 

.87611 

7.07167 

.89345 

8.38497 

53 

54 

.84184 

5.32279 

.85910 

6.09717 

.87640 

7.09052 

.89374 

8.41052 

54 

55 

.84213 

5.33429 

.85939 

6.11171 

.87669 

7.10946 

.89403 

8.43620 

55 

56 

.84242 

5.34584 

.85967 

6.12630 

.87698 

7.12849 

.89431 

8.46203 

56 

57 

.84270 

5.35743 

.85996 

6.14096 

.87726 

7.14760 

.89460 

8.48800 

57 

58 

.84299 

5.36906 

.86025 

6.15568 

.87755 

7.16681 

.89489 

8.51411 

58 

501  .84328 

5.38073 

.86054 

6.17046 

.87784 

7.18612 

.89518 

8.54037 

59 

60!  .84357 

5.39245 

.86083 

6.18530 

.87813 

7,20551 

.89547 

8.56677 

60 

580 


TABLE   XXII.— VERSINES  AND   EXSEOAXTS. 


/ 

84° 

85° 

86° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.89547 

8.56677 

.91284 

10.47371 

.93024 

13.33559 

0 

1 

.89576 

8.59332 

.91313 

10.51199 

.93053 

13.39547 

1 

2 

.89605 

8.62002 

.91342 

10.55052 

.93082 

13.45586 

2 

3 

.89634 

8.64687 

.91371 

10.58932 

.93111 

13.51676 

3 

4 

.89663 

8.67387 

.91400 

10.62837 

.93140 

13.57817 

4 

5 

.89692 

8.70103 

.91429 

10.66769 

.93169 

13.64011 

5 

6 

.89721 

8.72833 

.91458 

10.70728 

.93198 

13.70258 

6 

7 

.89750 

8.75579 

.91487 

10.74714 

.93227 

13.76558 

7 

8 

.89779 

8.78341 

.91516 

10.78727 

.93257 

13.82913 

8 

9 

.89808 

8.81119 

.91545 

10.82768 

.93286 

13.89323 

9 

10 

.89836 

8.83912 

.91574 

10.86837 

.93315 

13.95788 

10 

11 

.89865 

8.86722 

.91603 

10.90934 

.93344 

14.02310 

11 

12 

.89894 

8.89547 

.91632 

10.95060 

.93373 

14.08890 

12 

13 

.89923 

8.92389 

.91661 

10.99214 

.93402 

14.15527 

13 

14 

.89952 

8.95248 

.91690 

11.0339? 

.93431 

14.22223 

14 

15 

.89981 

8.98123 

.91719 

11.07610 

.93460 

14.28979 

15 

16 

.90010 

9.01015 

.91748 

11.11852 

.93489 

14.35795 

16 

17 

.90039 

9.03923 

.91777 

11.16125 

.93518 

14.42672 

17 

18 

.90068 

9.06849 

.91806 

11.20427 

.93547 

14.49611 

18 

19 

.90097 

9.09792 

.91835 

11.24761 

.93576 

14.56614 

19 

20 

.9C126 

9.12752 

.91864 

11.29125 

.93605 

14.63679 

20 

21 

.90155 

9.15730 

.91893 

11.33521 

.93634 

14.70810 

21 

22 

.90184 

9.18725 

.91922 

11.37948 

.93663 

14.78005 

22 

23 

.90213 

9.21739 

.91951 

11.42408 

.93692 

14.85268 

23 

24 

.90242 

9.24770 

.91980 

11.46900 

.93721 

14.92597 

24 

25 

.90271 

9.27819 

.92009 

11.51424 

.93750 

14.99995 

25 

26 

.90300 

9.30887 

.92038 

11.55982 

.93779 

15.07462 

26 

27 

.90329 

9.33973 

.92067 

11.60572 

.93808 

15.14999 

27 

28 

.90358 

9.37077 

.92096 

11.65197 

.93837 

15.22607 

28 

29 

.90386 

9.40201 

.92125 

11.69856 

.93866 

15.30287 

29 

30 

.90415 

9.43343 

.92154 

11.74550 

.93895 

15.38041 

30 

31 

.90444 

9.46505 

.92183 

11.79278 

.93924 

15.45869 

31 

32 

.90473 

9.49685 

.92212 

11.84042 

.93953 

15.53772 

32 

33 

.90502 

9.52886 

.92241 

11.88841 

.93982 

15.61751 

33 

34 

.90531 

9.56106 

.92270 

11.93677 

.94011 

15.69808 

34 

35 

.90560 

9.59346 

.92299 

11.98549 

.94040 

15.77944 

35 

36 

.90589 

9.62605 

.92328 

12.03458 

.94069 

15.86159 

36 

37 

.90618 

9.65885 

.92357 

12.08040 

.94098 

15.94456 

37 

38 

.90647 

9.69186 

.92386 

12.13388 

.94127 

16.02835 

38 

39 

.90676 

9.72507 

.92415 

12.18411 

.94156 

16.11297 

39 

40 

.90705 

9.75849 

.92444 

12.23472 

.94186 

16.19843 

40 

41 

.90734 

9.79212 

.92473 

12.28572 

.94215 

16.28476 

41 

42 

.90763 

9.82596 

.92502 

12.33712 

.94244 

16.37196 

42 

43 

.90792 

9.86001 

.92531 

12.38891 

.94273 

16.46005 

43 

44 

.90821 

9.89428 

.92560 

12.44112 

.94302 

16.54903 

44 

45 

.90850 

9.92877 

.92589 

12.49373 

.94331 

16.63893 

45 

46 

.90879 

9.96348 

.92618 

12.54676 

.94360 

16.72975 

46 

47 

.90908 

9.99841 

.92647 

12.60021 

.94389 

16.82152 

47 

48 

.90937 

10.03356 

.92676 

12.65408 

.94418 

16.91424 

48 

49 

.90966 

10.06894 

.92705 

12.70838 

.94447 

17.00794 

49 

50 

.90995 

10.10455 

.92734 

12.76312 

.94476 

17.10262 

50 

51 

.91024 

10.14039 

.92763 

12.81829 

.94505 

17.19830 

51 

52 

.91053 

10.17646 

.92792 

12.87391 

.94534 

17.29501 

52 

53 

.91082 

10.21277 

.92821 

12.92999 

.94563 

17.39274 

53 

54 

.91111 

10.24932 

.92850 

12.98651 

.94592 

17.49153 

54 

55 

.91140 

10.28610 

.92879 

13.04350 

.94621 

17.59139 

55 

50 

.91169 

10.32313 

.92908 

13.10096 

.94650 

17.69233 

56 

57 

.91197 

10.36040 

.92937 

13.15889 

.94679 

17.79438 

57 

58 

.91296 

10.39792 

.92966 

13.21730 

.94708 

17.89755 

58 

59 

.91255 

10.43569 

.92995 

13.27620 

.94737 

18.00185 

59 

60 

.91284 

10.47371 

.93024 

13.33559  1 

.94766 

18.10732 

60 

TABLE  XXII.-VERSlNES  AND  EXBEOANTS. 


381 


/ 

87° 

88° 

89° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.94766 

18.10732 

.96510 

27.65371 

.98255 

56.29869 

0 

1 

.94795 

18.21397 

.96539 

27.89440 

.98284 

57.26976 

1 

2 

.94825 

18.32182 

.96568 

28.13917 

.98313 

58.27431 

2 

3 

.94854 

18.43088 

.96597 

28.38812 

.98342 

59.31411 

3 

4 

.94883 

18.54119 

.96626 

28.64137 

.98371 

60.39105 

4 

5 

.94912 

18.65275 

.96655 

28.89903 

.98400 

61.50715 

5 

6 

.94941 

18.76560 

.96684 

29.16120 

.98429 

62.66460 

6 

.94970 

18.87976 

.96714 

29.42802 

.98458 

63.86572 

7 

8 

.94999 

18.99524 

.96743 

29.69960 

.98487 

65.11304 

8 

9 

.95028 

19.11208 

.96772 

29.97607 

.98517 

66.40927 

9 

10 

.95057 

19.23028 

.96801 

30.25758 

.98546 

67.75736 

10 

11 

.95086 

19.34989 

.96830 

30.54425 

.98575 

69.16047 

11 

12 

.95115 

19.47093 

.96859 

30.83623 

.98604 

70.62285 

12 

13 

.95144 

19.59341 

.96888 

31.13366 

.98633 

72.14583 

13 

14 

.95173 

19.71737 

.96917 

31.43671 

.98662 

73.73586 

14 

15 

.95202 

19.84283 

.96946 

31.74554 

.98691 

75.39655 

15 

16 

.95231 

19.96982 

.96975 

32.06030 

.98720 

77.13274 

16 

17 

.95260 

20.09838 

.97004 

32.38118 

.98749 

78.94968 

17 

18 

.95289 

20.22852 

.97033 

32.70835 

.98778 

80.85315 

18 

19 

.95318 

20.36027 

.97062 

33.04199 

.98807 

82.84947 

19 

20 

.95347 

20.49368 

.97092 

33.38232 

.98836 

84.94561 

20 

21 

.95377 

20.62876 

.97121 

33.72952 

.98866 

87.14924 

21 

22 

.95406 

20.76555 

.97150 

34.08380 

.98895 

89.46886 

22 

23 

.95435 

20.90409 

.97179 

34.44539 

.98924 

91.91387 

23 

24 

.95464 

21.04440 

.97208 

34.81452 

.98953 

94.49471 

24 

25 

.95493 

21.18653 

.97237 

35.19141 

.98982 

97.22303 

25 

26 

.95522 

21.33050 

.97266 

35.57633 

.99011 

100.1119 

26 

27 

.95551 

21.47635 

.97295 

35.96953 

.99040 

103.1757 

27 

28 

.95580 

21.62413 

.97324 

36.37127 

.99069 

106.4311 

28 

29 

.95609 

21.77386 

.97353 

36.78185 

.99098 

109.8966 

29 

30 

.95638 

21.92559 

.97382 

37.20155 

.99127 

113.5930 

30 

31 

.95667 

22.07935 

.97411 

37.63068 

.99156 

117.5444 

31 

32 

.95696 

22.23520 

.97440 

38.06957 

.99186 

121.7780 

32 

33 

.95725 

22.39316 

.97470 

38.51855 

.99215 

126.3253 

33 

34 

.95754 

22.55329 

.97499 

38.97797 

.99244 

131.2223 

34 

35 

.95783 

22.71563 

.97528 

39.44820 

.99273 

136.5111 

35 

36 

.95812 

22.88022 

.97557 

39.92963 

.99302 

142.2406 

36 

37 

.95842 

23.04712 

.97586 

40.42266 

.99331 

148.4684 

37 

38 

.95871 

23.2163? 

.97615 

40.92772 

.99360 

155.2623 

38 

39 

.95900 

23.38802 

.97644 

41.4452E 

.99389 

162.7033 

39 

40 

.95929 

23.56212 

.97673 

41.97571 

.99418 

170.8883 

40 

41 

.95958 

23.73873 

.97702 

42.51961 

.99447 

179.9350 

41 

42 

.95987 

23.91790 

.97731 

43.07746 

.99476 

189.9868 

42 

43 

.96016 

24.09969 

.97760 

43.64980 

.99505 

201.2212 

43 

44 

.96045 

24.28414 

.97789 

44.23720 

.99535 

213.8600 

44 

45 

.96074 

24.47134 

.97819 

44.84026 

.99564 

228.1839 

45 

46 

.96103 

24.66132 

.97848 

45.45963 

.99593 

244.5540 

46 

47 

.96132 

24.85417 

.97877 

46.09596 

.99622 

263.4427 

47 

48 

.96161 

25.04994 

.97906 

46.74997 

.99651 

285.4795 

48 

49 

.96190 

25.24869 

.97935 

47.42241 

.99680 

311.5230 

49 

50 

.96219 

25.45051 

.97964 

48.11406 

.99709 

342.7752 

50 

51 

.9G248 

25.65546 

.97993 

48.82576 

.99738 

380.9723 

51 

52 

.%277 

25.86360 

.98022 

49.55840 

.99767 

428.7187 

52 

53 

.96307 

26.07503 

.98051 

50.31290 

.99796 

490.1070 

53 

54 

.96336 

26.28981 

.98080 

51.09027 

.99825 

571.9581 

54 

55 

.96365 

26.50804 

.98109 

51.89156 

.99855 

686.5496 

55 

56 

.96394 

26.72978 

.98138 

52.71790 

.99884 

858.4369 

56 

57 

.96423 

26.95513 

.98168 

53.57046 

.99913 

1144.916 

57 

58 

.96452 

27.18417 

.98197 

54.45053 

.99942 

1717.874 

58 

59 

.96481 

27.41700 

.98226 

55.35946 

.99971 

3436.747 

59 

60 

.96510 

27.65371 

.98255 

56.29869 

1.00000 

Infinite 

60 

382 


TABLE  XXIII.— USEFUL  NUMBERS  AND  FORMULAS. 


Ratio  of  circumference  to  diameter n  3.1415906530 

Reciprocal  of  same .3183098862 

/T 

4/iT  1.7724 538509 

rr2  9.8696044011 

Degrees  in  arc  equal  to  radius 1§2  57  295779513 

IT 

Minutes  in  arc  equal  to  radius -      -    3437.74677078 

I  Seconds  in  arc  equal  to  radius —      -  206264 .80024 

'  Length  of  seconds  pendulum  at  New  York  in  feet 3.25938 

Square  root  of  same 1.8054 

I  Acceleration  due  to  gravity  at  New  York g  32.1688 

Square  root  of  same ^g  5.67175 

Cubic  inches  in  U.  S.  gallon 231 

Cubic  inches  in  Imperial  gallon 277.274 

Cubic  inches  in  U.  S.  bushel 2150.42 

Cubic  feet  in  U.  S.  bushel    1.244456 

U.  S.  gallons  in  one  cubic  foot  7.4805 

I iirperial  gallons  in  one  cubic  foot 6.2321 

AUvsJmum  weight  of  one  cubic  foot  of  water  (%  100  nearly)- •  62.4 

:\  ii'titoer  of  gr.iins  in  one  pound  avoirdupois 7000 

Number  of  grains  in  one  ounce  avoirdupois 437.5 

Number  of  grains  in  one  pound  troy 5760 

Number  of  grains  in  one  ounce  troy 480 

Circumference  of  circle  (radius  r) 2nr 

Area  of  circle  (radius  r) 7r>2 

Area  of  sector  (arc  of  a  degrees) ~ -Trr2 

oOO 

Area  of  sector  (length  of  arc  /) 

Area  of  segment  (chord  =  c,  middle  ordinate  =  m),  nearly.. 

Surface  of  a  sphere  (diameter  d) 

Volume  of  a  sphere  (diameter  d) 

Area  of  a  triangle  each  side  unity  .4330 

Area  of  a  pentagon  each  side  unity 1 . 7205 

Area  of  a  hexagon  each  side  unity'  =  .4330  X  6  =  2.5980 

Area  of  an  octagon  each  side  unity  4.8284 

Volume  spherical  segment  of  one  base  of  radius  r  and  altitude  a  is 

=  ^irar2  -f  %na3  =  1.5708ara  +  .5236a3; 

=  1.58a?-'2  -j-  .5a3,    very  nearly; 

=  1.6ar3  +  .5a3,    nearly. 

L^t  A  and  a  represent  the  bases  of  a  frustum  of  a  pyramid,  L  and  I  the 
lengths  of  corresponding  sides,  h  the  altitude  and  al  the  area  of  either 
base  for  side  equal  to  unity. 

Then   volume  =  ^  (L*  +  Z2  +  IL). 

o 

This  formula  is  immensely  shorter  and  better  than  that  given  in  the 
books,  namely,  vol  =  —(A -\~  a  -\-  \ Aa),  since  this  latter  requires  extra 

and  needless  computation  in  each  term,  besides  the  extraction  of  the 
square  root. 

!  When  I  =  0,    we  get  volume  of  pyramid  =  —  ; — . 

o 

i  Let  R  and  r  represent  the  radii  of  a  frustum  of  a  cone,  and  h  its  altitude. 
Then  volume  =  ^nh(R^  +  r2  +  rR). 
If  r  —  0  we  have,    volume  of  cone  =  J^/ijTrR3. 


TABLE  XXIV. 


383 


CONVERSION  OF  ENGLISH  FEET  INTO  METRES. 

6 

Feet. 

0 

1 

2 

8 

4 

5 

7 

8 

9 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

0 

0.000 

0.3048 

0.609( 

5    0,9144 

1.2192 

1.5239 

1.82 

872.1335 

2.4383 

2.7431 

10 

3  0479 

3.3527 

3.  6575 

)    3.9623 

4.2671 

4.5719 

4.87 

6715.1815 

5.4863 

5.7911 

20 

6.0959 

6.4006 

6.705J 

>    7,0102 

7.3150 

7.6198 

7.92468.2294 

8.5342 

8.8390 

30 

9.1438 

9.4486 

9.753^ 

[    10.058 

10.363 

10.668 

10.9 

7211.277 

11.582 

11.887 

40 

12.192 

12.496 

12.80 

13.106 

13.411 

13.716 

14.  C 

2014.325 

14.630 

14.935 

50 

15.239 

15.544 

15.841 

)    16.154 

16.459 

16.763  17.068  17.373 

17.678 

17.983 

60 

18.287 

18.592 

18.89' 

"    19.202 

19.507 

19.811 

20.11620.421 

20.726 

21.031 

70 

21.335 

21.640 

21.94. 

)    22.250 

22.555 

22.859 

23.1 

6423.46t) 

23.774 

24.079 

80 

24.383 

24.688 

24.99- 

J    25.298 

25.602    25.907 

86.5 

1226.517 

26.822 

27.126 

90 
100 

27.431 
30.479 

27.736 

30.7S4 

28.04 
31.08< 

28.346 
)    31.394 

28  .  651    28  .  955  29  .  260  29  .  565 
31.6981  32.003132.30832.613 

29.87(1 
32.918 

30.174 
33.222 

CONVERSION  OF  METRES  INTO  ENGLISH  FEET. 

Met, 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Feet. 

Feet. 

Feet.     Feet. 

Feet. 

Feet. 

Feet. 

Feet 

Feet. 

Feet. 

0 

0.000 

3.2809 

6.5018    9.8427 

13.123 

16.404 

19.685 

22.966 

26.247 

29.528 

10 

32.809 

36.090 

39.371    42.651 

45.932 

49.213 

52.4 

94 

55.77E 

59.056 

62.337 

20 

65.618 

68.899 

72.179    75.461 

78.741 

82.02-2 

85.303 

88.584 

91.865 

95.146 

30 

98.427 

101.71 

104.99    108.27 

111.55 

114.83 

118 

11 

121.3S 

124.6" 

127.96 

40 

131.24 

134.52 

137.80    141.08 

144.36 

147.64 

150 

9-2 

154.2C 

157.48 

160.76 

50 

164.04 

167.33 

170  61    173.89 

177.17 

180.45183.73 

187.01 

190.29 

193.57 

60 

196.85 

200.13 

203.42    206.70 

209.98 

213.26 

210 

r,4 

219.82 

223.10 

226.38 

70 

229.66 

232.94 

236.22    239.51 

242.79 

246.07249. 

3r. 

252.69 

255.91 

259.19 

80 

262.47 

265.75 

269.03    272.31 

275.60 

278.88 

•js-j. 

1(5 

285.44 

288.72 

292.00 

00 

295.28 

298.56 

391.84    305.12 

308.40 

311.69314. 

97 

318.25 

321.53 

324.81 

ing 

328.09    331.37 

334.65    337.93 

341.21 

344.49347.78 

351.06 

354.34 

357.62 

CONVERSION  OF  ENGLISH  STATUTE-MILES 

INTO  KILOMETRES. 

Miles. 

0 

1 

2 

3 

4 

5 

6 

7            8 

9 

Kilo.    Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo.     Kilo. 

Kilo. 

0 

0.0000  1.60933.2186 

t.8279 

6.4372    8.0465 

9.6558 

11.265212.8745 

14.4848 

10 

16.  093  117.702  19.  312 

20.921 

22.530    24.139 

25 

74!) 

27.358    28.967 

30.577 

20 

32.18633.79535.405 

J7.014 

38.6-23 

40.232 

41 

.842 

43.451    45.060 

46.670 

30       48.27949.88851.498 

33.107 

54.716 

56.325 

57 

.9351  59.544   61.153 

62.763 

40       64.37265.98167.591 

59.200 

70.8091  72.418 

74 

.028 

75.637    <7.246 

78.856 

50      i  80.  465  82.  074  83.  684 

35.293 

86.902 

88.511 

90 

.121 

91  730    93.339 

94.949 

60      '96.55898.16799.777 

101.39 

102.99 

104.60 

10 

(3.21 

107.82    109.43 

111.04 

70 

112.65ill4.26115.87 

117.48 

119.08 

120.69 

122.30 

123.  91j  125.52 

127.13 

80 

128.74  130.35131.96 

133.57 

135.17 

136.78 

13 

8.89 

140.00    141.61 

143.22 

90 

144.  851146.  44  148.05 

149.66 

151.26 

152.87 

154.48 

156.09    157.70 

159.31 

:    100 

160.93162.53164  14 

165  75 

167.35 

168.96 

170.57 

172.181  173.79 

175.40 

[CONVERSION  OF  KILOMETRES  INTO  ENGLISH  STATUTE-MILES 

!  Kilom. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0 

Miles.  Miles. 
O.OOOOiO.6214 

Miles. 
1.2427 

Miles. 
1.864! 

Miles. 

2.4855 

Miles. 

3.1069 

Miles. 
3.7282 

Miles. 
4.3497 

Miles' 
4.9711 

Miles. 
5.59?4 

10 

6.213b 

6.8352 

7.4565 

8.0780 

8.6994 

9.3208 

9.94211  10.562 

11.185 

11.805 

20 

12.427 

13.049 

13.670 

14.292 

14.913 

15.534 

16 

.156 

16.776 

17.399 

18.019 

30 

18.641 

19.263 

19.884 

20.506 

21.127 

21.748 

23 

370 

22.990 

23.613 

24.233 

1       40 

24.85f 

25.477 

26.098 

26.72027.341 

27.962 

28.584|  29.204 

29.827' 

30.-J47 

50 

31.06< 

31.690 

32.311 

32.033J33.554 

34.175 

34.7971  35.417 

36.040 

36  660 

60 

37.28:; 

37.904 

38.525 

39.147 

39.768 

40.3891  41 

.011 

41.631 

42.  254 

42.874 

70 

43.49' 

44.118 

44.739 

45.361 

45.982 

46.603    47 

47.845 

iS.468 

49.088 

80 

49.711 

50.332 

50.953 

51  575 

52  196 

52.8171  53.4391  54.059 

54.682 

55.302 

90 

55.924 

56.545 

57.166 

57.78858.400 

59  050 

58 

.  o.Y, 

30.272 

60.895 

61.515 

100 

62.  13* 

!6°,.759 

63.380 

64.002164.623 

35.244 

(55.866 

!  66.486 

67.109 

67.729 

386  INDEX. 


PAGE 

Curves,  how  to  lay  out  on  the  ground: 
With  transit  and   chain  ...............................................     53 

With    chain    only  ......................................................     §- 

Curves,   slackening   speed   on  ...........................................  nj 

Should    be    flattened    on    grades  ......................................     n 

Middle  ordinates,  Table  V,  and  ordinates  to  short  chords,  approxi- 


mate values. 


51 

Problems   on   curves 65 

To  locate  curve  when  vertex  is  inaccessible 87 

To  locate  when  the  vertex  and  the  ends  are  inaccessible 87 

Problems  applicable  to  passing  obstacles  generally 65,  74,  88 

Curves,  compound,  formulas  for 115,  178 

Radii  and  tangents  compared 119 

Problems    pertaining    to 120,  150 

Problems,    special 124 

General  relations  deduced  by  substitutions 145 

Curves   tangent  to   curves,   problems   on 151 

To     find     tangents     of    a     compound     curve     of    any     number     of 

branches    176 

Wye   problems 165,  170 

'Curves,    concentric,    problems    concerning 170,  173 

Curves,  distance  apart  starting  from  same  point,  etc 175 

To  locate  the  second  branch  of  a  compound  curve  from  a  point  on 

the    first   branch 174 

Curves,  reversed,  problems  on 184 

Curves,    turnout,    difference    between    ending    on    tangents    and    con- 
necting   with    tracks 199 

Curve,  the  true  transition: 

Definitions   and  essential  requisites x?3 

Elementary   relations 

To  lay  out  the   curve 

Special  problems  and  laying  out  the  curve 

To  lay  out  the  curve  from  different  points  on  it 245 

Comparison    of    the    true    curve    with    the    compound    curve    com- 
monly used  as  such 151 

Curves,   vertical 107 

Curved  tracks,  angle  of  intersection 178 

Datum    in    leveling 90 

Deflection    angles 53 

Degree  of  curve 42 

Distances  of  frogs  from  switch,  Table   VI 187 

Earthwork: 

Area   of   level    section 254 

Area  of  section  not  level 255 

Formulas    for   volumes 256 

"End-area  volume^"  and  "middle-area  volumes"   compared 257 

Special    formulas    and    cases 259 

Loaded  flat  cars,  piles   of  broken   stone,   etc 263 

Ends   of  embankments   or   "dumps" 264 

Ground   irregular  laterally , -'64 


INDEX.  387 

PAGE 

Karthwork : 

Mixed    work 265 

Correction   for   curvature 267 

Overhaul    268 

Monthly    estimates 269 

Final    estimates 270 

Computation    of   volume   of   prismoids   level   laterally 270 

Clearing    and    grubbing 284 

Staking   out    work 286 

Borrow-pits    294 

Shrinkage    294 

Two  important  principles  in   ''staking  out" 291 

Elevation   a  relative  question 90 

Elevation    of    outer    rail 112 

Simple    and    accurate   formula   for   elevation 113 

Proper    elevation    to    use 113 

1'ield-book,    form    of,    for    leveling .' 92 

Frogs   and   switches,   Table   VI 318 

Grade    line -'84 

Gradienter,  formulas  deduced   for   it   require  no   computation 105 

Horizon.      To    find    height    of    object    by    the    dip    of    the    horizon. 

Special   and   exact  formulas   apply  without  computation 98 

Level,   to   adjust 18 

Use  and  care  of 22 

Leveling     , 90 

How  to  keep  notes  on 92 

Proof  of  correctness 93 

Trigonometric    97 

Locate  a  level   line,   to 94 

Locate  a  grade  line,  to.... 9; 

Location     5 

Locating   engineer,    qualifications    for 9 

Maps  to  be  used  freely  in  studying  the  country 7 

Needle,    magnetic,   to  adjust 23 

How  to  use 23 

Numbers   forming   integral    sides   of  right   triangles 76 

Obstacles  ,in   surveying 76 

To   erect   a   perpendicular 76 

To  let  fall   a  perpendicular 77 

Ditto   from    an   inaccessible    point 77 

To  prolong  a  line 78 

Obstacles  to  measurement  of  line: 

When   one   end   is   inaccessible 79 

When  both  ends  of  the  line  are  inaccessible 80 

A  line  perpendicular  to  an  inaccessible   line 81 

A  line  determined  from  another  line  across  an  inaccessible  space.  81 

Offsets,    to    calculate 48 

Preliminary    survey 5,  82 

Plane   right  triangles,    solution   of 3t 

Plane  oblique  triangles,   solution   of ...,,... , 35 


388  INDEX. 


PAGE 

Piers,   bridge 298 

Reconnoissance    c 

Pocket   compass   for 8 

Requirements   for g 

Resistance,   train 9 

Sines,    tangents    and    secants    defined 22 

Slope  stakes,  to   set 286 

Stadia.     Formulas   deduced   and   simplified 101 

Streams,  bends  in,  to  avoid 7 

Surveys,    preliminary 5,  82 

"Tangent  proportion"  deduced 33 

Tangent  to  a  curve  from  fixed  point,  how  to  locate 75 

Tangent  of  any  compound  curve,  how  to  find  the  length 176 

Tables.     See  Contents. 

Track-laying    296 

Transit,    to    adjust 12 

Use  and  care  of 15 

Trigonometry    25 

The  application  of  but  one  geometrical  principle 27 

Tunnels 298 

Turnouts: 

Single    196,  212,  216 

Double 208,  210,  214,  217,  219 

Traversing    82 


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Byrne,  C.E.  8vo,  cloth.  3d  edition,  revised  and  greatly 
enlarged 5.  ou 

THE  TRANSITION  CQRVE 

By  Professor  Charles  L.  Crandall,    16mo,  morocco  flap 1  50 


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